From a Trinitarian perspective, does the term 'only begotten Son' make sense outside of the concept of the incarnation?

The Old Roman Creed hints of a perspective that might address your question:

I BELIEVE in God almighty, the Father almighty and in Christ Jesus, his only Son, our Lord who was born of the Holy Spirit and the Virgin Mary who was crucified under Pontius Pilate and was buried and the third day rose from the dead who ascended into heaven and sitteth on the right hand of the Father whence he cometh to judge the living and the dead. And in the Holy Spirit, the holy church, the remission of sins, the resurrection of the flesh, the life everlasting.

Notice that Jesus incarnate was born of the Holy Spirit and Mary, the Son of God (only begotten)and the "Son of Man" born of the overshadowing presence of the Holy Spirit with Mary.

The Moravians & Count Zinzendorf prior to the Reformation held that the Holy Spirit was God the Mother. In this the Holy Spirit eternally proceeds from God the Father bearing His eternal will. The manifest will of the Father born by the Holy Spirit then is Eternally Begotten. Jesus Christ then is the manifest will of God the Father born of the Holy Spirit. This in extension would also hold that the Holy Spirit bear's out Christ in the believer’s life.

A differentiation between the Holy Spirit’s procession from the Father and the Son’s sending of the Holy Spirit must be made. The sending by the son is as a result of His finished work on the cross.

In this case Jesus is the Eternally Begotten!

Begetting ( present participle of the verb "beget") - To procreate an offspring. To beget is to make someone have one's nature.

The purpose of begetting is to pass on genome ( whole nature) to the offspring.

Elements of Divine Begetting

1) Male ( John 1:14,18)

2) Womb ( Ps. 110:3, John 1:18)

53) Divine nature (John 1:1,18)

Elements of Human Begetting

1) Male

2) Female

3) Intercourse

4) Womb

5) Human nature (genome)

As we can clearly see, the divine begetting is an inscrutable mystery because there is no Biblical information about How the Father produced the Son.

Anyway, the 'how' of the divine begetting is not what's important but rather, what's important is its essence (the meaning it conveys) which is co-equality of the Father and the Son in one nature.

The Scriptures only speaks of an analogy from psychology in regards to the Son's begetting from the Father.

Word from Mind ( Psalm 45:1, John 1:1,14,18).

Wisdom from Mind ( Proverbs 8:22, John 1:1,14,18).

The Mind of God the Father eternally exists as rational and wise.Therefore, the Son of God the Father eternally exists.

Only begotten from the Father (Psalm 109:3, John 1:14).

The psychological analogy fails to explain how the Son ( a person) is produced by the Father ( a person). The psychological analogy only shows that the production is necessary and eternally exists.In fact, this analogy has its own inscrutable mystery too because we do not know How God produced both Word and Wisdom from His mind without a beginning of its existence.

Produced not from nothing but from the Father

Word = produced ( Ps. 45:1 LXX)

Wisdom = produced ( Prov. 8:22 LXX)

Son = produced ( John 1:14)

Eternal ( Without beginning or ending)

Word = eternal ( John 1:1)

Wisdom = eternal ( 1 Timothy 1:17)

Son = eternal ( Micah 5:2, John 1:1,18, Colossians 2:9)

The Scriptures do not reveal how these were produced. The fact that these are eternally existent and simultaneously productions of God, is a mind blowing reality.

If any one, therefore, says to us, How then was the Son produced by the Father? we reply to him, that no man understands that production, or generation, or calling, or revelation, or by whatever name one may describe His generation, which is in fact altogether indescribable. Neither Valentinus, nor Marcion, nor Saturninus, nor Basilides, nor angels, nor archangels, nor principalities, nor powers [possess this knowledge], but the Father only who begot, and the Son who was begotten (Against Heresies 2.28.6)

One other possibility to consider, and for me this is the preferred options, is that “begotten” is not really the right word to use at all.

The Greek word we are discussing here is “monogenes” [μονογενεσ]. Of course this is a compound word, and the first half is easy, “mono" meaning “one”. The problem comes in the second half, as there is no Greek word exactly “genes”, so originally translators decided it must be a form of the word “gennao”, which means to “beget", “bear" or "give birth to".

However, it appears “genes” might not be a derivative of the verb “gennao” but rather the noun “genos”, which means “kind", “class" or “ family". It’s where we get the words "gene" and "genre" from. Thus “monogenes” means “one of a kind”, “unique” or “the only one in it’s class”. You could say a Platypus is “monogenes”, it’s the only one of it’s kind, there’s nothing else like it.

We get further confirmation for this in Hebrews 11:17 where Isaac is called Abraham's "monogenes" son. We know that Isaac was not the “only begotten" son of Abraham, there was the older Ishmael as well as six other sons by a concubine named Keturah. But Isaac was the only "son of promise", so in that way he was a unique, one of a kind son.

You may have noticed that most of the modern translations have moved away from “begotten” in John 3:16. The NIV and HCSB render it "one and only Son" and the ESV and CEV say "only Son".

What is also interesting to me is that the creed writers seemed to be uncomfortable with the word "begotten" because they almost immediately feel the need to further define it by saying "begotten, not made".

Aquinas addresses the question from different angles a number of places in the Summa Theologica, in various parts of the "Treatise on the Most Holy Trinity" (First Part, Questions 27–43).

The fundamental question is answered more or less directly in Question 27, "The Procession of the Divine Persons". Article 2 of this question, "Whether any procession in God can be called generation?", talks about what it might mean to say that the Son "is begotten by" the Father. Thomas' answer is:

... We must observe that generation has a twofold meaning: one common to everything subject to generation and corruption; in which sense generation is nothing but change from non-existence to existence. In another sense it is proper and belongs to living things; in which sense it signifies the origin of a living being from a conjoined living principle; and this is properly called birth.

Not everything of that kind, however, is called begotten; but, strictly speaking, only what proceeds by way of similitude. ... the procession of the Word in God is generation; for He proceeds by way of intelligible action, which is a vital operation:—from a conjoined principle (as above described):—by way of similitude, inasmuch as the concept of the intellect is a likeness of the object conceived:—and exists in the same nature, because in God the act of understanding and His existence are the same, as shown above. Hence the procession of the Word in God is called generation; and the Word Himself proceeding is called the Son.

The Son, in other words, is "begotten" not in virtue of the Incarnation, but because the Son is the Word of God proceeding from, and in that sense "begotten by" (one could say "generated by"), the Father.

There are a few other places where Aquinas discusses different aspects of the question (such as "Is it appropriate to call the Father 'unbegotten'?"), but this is the clearest exposition of Aquinas' actual position, and it appears from his discussions as a whole to be a standard Trinitarian answer.