Why must the hypostatic union be viewed as taking place at Jesus' conception?

I'm sorry if I misunderstand your "scoping the question", but I'm responding from the Protestant tradition which I think has something significant to say on the matter.

In Reformed theology, the implications for Jesus' hypostatic union beginning at his conception at the least encompass his life of obedience. Jesus actually was born and lived under the law (see Galatians 4:4-5) and kept that law as the second Adam. This law keeping would not have been possible without him being perfectly upheld by the Spirit. If he had not been upheld by the Spirit he could have "theoretically" fallen from his original righteousness. However, this could never have happened because he actually was upheld by the Spirit.

Jesus needed to be both God and man from conception so that he could live his whole life in obedience to his Father. From at least his circumcision onward to the cross, his life was one of suffering and enduring temptation without giving in. He was perfected through his sufferings though he was already sinless. In Hebrews 9:14, Jesus' quality of being an unblemished sacrifice is attributed to the eternal Spirit. In other words, his whole life of obedience which qualified him to be an unblemished sacrifice was due to the Spirit's work throughout his whole life. The obedience is also what earned him righteousness which he then credits to those who have faith in him. So it's fair to say that the Reformed view of the hypostatic union requires it to begin at conception or there is no salvation possible.

Why must the hypostatic union be viewed as taking place at Jesus' conception?

Because the man assumed by the Logos is His humanity and no one else. The man belongs to Logos. If the union happens at baptism then by nature that man is not divine but merely endowed with grace of adoption to be one of His children.

One Lord Jesus Christ, the only-begotten Son of God, begotten of the Father before all ages. ... Who for us men and for our salvation came down from heaven and was incarnate of the Holy Spirit and the Virgin Mary and became man.

Niceano-Constantinopolitan Creed (381)

The Logos became man for us and for our salvation. He took humanity from the Theotokos this is why devotion to her is crucial for proper Christology. She is the birth giver of God the Word. She gave her humanity to the Logos. Without being born of a virgin the Logos must create His own humanity alien from the rest of humanity because His humanity would be newly created and not identical with the fallen nature like the rest of us.

If any one confess not that Emmanuel is in truth God and that the holy Virgin is therefore Mother of God (Theotokos), for she bare after the flesh the Word of God made flesh, be he anathema.

St. Cyril, Twelve Anathemas, 1.

God the Word won't create a fallen humanity for Him to assume because He is not the author of evil or partake in evil. It's crucial for the Logos to assume our nature to redeem us and that is why He was united with our fallen humanity at conception.

What has not been assumed has not been healed

St. Gregory of Nazianzus, Epistle 101:32.

As regards to whether or not the Holy Spirit guarded Jesus from falling into sin, it's worth to note that even though Jesus is filled with the Spirit in His earthly ministry He is incapable to sin because He is a divine person, not divine-humane person.

If any one say that the one Lord Jesus Christ hath been glorified by the Spirit, using His power as though it were another's, and from Him receiving the power of working against unclean spirits and of accomplishing divine signs upon men; and does not rather say that His own is the Spirit, through whom He has wrought the divine signs, be he anathema.

St. Cyril, Twelve Anathemas, 9.

Because our fallen humanity was assumed at the moment of conception the Logos is one of us. If He assumed the man Jesus of Nazareth at later time and not at Jesus' conception then when He assumed Jesus there won't be real union but mere Samosatan adoptionism. By adoptionism the Logos by nature is not identical with us and our nature won't be healed. Only by assuming the man at conception He assumed humanity to Himself and it belongs to Him not someone else. By redeeming that humanity He redeemed us.

It's really very simple, and it's all about identity.

We acknowledge that it is "the Word" who was "made flesh" because Scripture says: "the Word was made flesh and dwelt among us" (Jn 1:1; 14)—not absorbed, added to, or confused with flesh. So there is only one Person of which to speak: the Word, called Jesus since the Incarnation. No 'human Jesus' (as if there is a human and a divine Jesus, instead of one Person with two natures equally His) which was 'mixed with' the Word. Jesus is the Word, the Son of God in the flesh: "and we beheld his glory: the glory of the Only-Begotten of God, full of grace and truth."

So the reason for placing the hypostatic union at the conception of Jesus is made obvious: the "Son" whom Mary was to "conceive and bear" (Lk 1:31) is the Word, the Son of God (Gal 4:4)—ever since there was flesh to call Jesus', it belonged to the Son. The humanity of Jesus is and has been His own since it began to be formed in Mary (Lk 1:42-43—notice the fruit of Mary's womb is the Lord's own already; cf. Mt 2:20).

There are not two people in Jesus, but two natures belonging to the one Person: the Word/Son of God.

This question is asked with regard to "Trinitarian orthodoxy", which is where a bit of history regarding the early centuries of the Church reveals a lot as to the matter of beliefs about the two natures of Christ - his deity, and his manhood. The question of when this might have happened is related to this hypostatic union.

Orthodox teaching is that the Trinity means one substance and three persons; the hypostases (persons) of the Trinity share a common ousia (nature). Nestorius (b. after 381) stood against the Arian heresy and supported his chaplain Anastasius who objected to the popular designation of Mary as 'bearer' of God. He taught that the two phyesis (natures) of Christ have different proposa (persons) attached. For Nestorius, the union that makes up Christ has the divine one as eternal and omnipotent, while the human one is mortal and weak. Such 'union of natures' cannot be as strong as that between the persons of the Godhead. He did not want the humanity of Jesus to be swallowed up in the divinity or truncated or in any way made less than like ours. In spite of his attempts to explain how a conjunction of two persons could count as one person (proposon), his Christ turned out to be two individuals and not one. The Son of God did not truly experience human existence "in the flesh" but only through association with the man. Nestorius's Christology was little more than warmed-over and dressed-up adoptionism, as Cyril of Alexandria pointed out (412-44). This is where delving into the teaching of earlier adoptionism will show this clash with orthodox trinitarian teaching. Cyril's unique contribution to Christology is the doctrine of the hypostatic union, in its basic outlines. (Source: The Story of Christian Theology by Roger E. Olson, pp 217-219)

This gives some sketchy background to the Athanasian Creed of the late 5th century. Orthodox teaching with regard to the duality of the nature possessed uniquely by Jesus of Nazareth, is that these two natures cannot 'merge' or 'mingle'. They are two different things that unite only in the Person of Jesus Christ. Christ's divine and human natures do nor merge or mingle. Both natures meet and unite only in the Person of Christ.

Therefore, the question as to when this happened is linked to why Mary had to be a virgin. The 'conception' is the joining of that eternal begetting with that which is 'holy', by the Holy Spirit (Matthew 1:20). The Word became flesh at that point, for it was Mary who provided human nature, the One conceived in her womb being fully human, sinless, yet in no way devoid of his eternally existing divine nature. Those two natures met and united at the start of conception, by the power of the Holy Spirit. Those who deny the pre-human existence of this Holy One in Mary's womb have no choice but to alight on some external event that they can claim as a point in time when a human nature became infused with 'something more' from above, and the events at Jordan's banks suit their ideas.

Some claim this only happened at the mature Jesus' water baptism, but you will find that many who claim this do not believe the orthodox Trinity doctrine. Many claim that this Jesus was no more than a sinless, perfect man. However, the gospel accounts tell us that at the point of his birth, he was already a Saviour, "Christ the Lord" - Luke 2:11.

It is, quite simply, denial of the Word of God being God, in the beginning, so that this Word made everything that was made, that leads some to think it must have been the baptism of the man, Jesus, that marked a significant point of God's approval and divine authority and enabling. Yet at his baptism, the Father declared that he was already well pleased with his Son, before his ministry had even started! Mark 1:11 has God speaking in the past tense, "Thou art my Son, the Beloved, in whom I did delight" (NLT).

So, when you ask, "Why must the hypostatic union be viewed as taking place at Jesus' conception?" it is because of who this One already was, prior to the virgin conceiving, and the Holy Spirit's role in that miraculous conception which brought human nature to meet and unite with the One who made everything that was made (John 1:1-14).

The doctrine of the Trinity says that God is three persons in one being. One of those persons, the second person of the Trinity, took on flesh and became a human, the man Jesus.

If I understand what you're suggesting, you would run into one of these two problems:

1. Because the pre-baptism Jesus was a person, the hypostatic union would be two people, a heresy called Nestorianism. This is a very complicated topic, so I'll quote from a blog I found that explains some of the problems much better than I could:

   When pressed the Nestorian will say that Jesus is both God and man but what he means by this is that Jesus is one person and God is another person and two persons reside in the same human body. In other words, Jesus is not really God at all. God is somehow in Jesus but not really Jesus. Therefore, the logical and rational conclusion to be drawn is that Jesus is not fully God and it would therefore be idolatry to worship Jesus.
   ...
   The Bible never speaks of Jesus as if He were two persons. Rather the Bible speaks of Jesus as though He were one individual man who is as fully human as anyone else. He thinks, feels, eats, acts, lives and dies just as any other human person would. But Jesus is more than a human person. He is also a divine person and this person is one person being both divine and human at one and the same time. The other error of Nestorianism is that it denies that the Son of God, the Word, became flesh.
   ...
   The Nestorian thinks he has solved the problem of the apparent contradiction of Jesus being one person who is both fully God and fully human. But the Nestorian only thinks he has solved the "apparent" problem which is no real problem at all. And even more than that the Nestorian position divides Christ into two persons which are not united at all.
   
   (by Charlie J. Ray, emphasis added)

   If the Word is a different person than the human Jesus then really the whole idea of the incarnation is lost.

2. Something that is not God became God. This is very similar to the heresy called Adoptionism, as some people who believe that say that Jesus was adopted as God's son and became divine. This goes against God's singular uniqueness of being the eternal God. This is much less controversial (I think the only 'Christian' group which says that the non-divine can become divine is the LDS) so I'll leave it there.