How could Christ offer His blood to those under the Law, when He was forbidden by heritage?

The following is a direct quote from Barnes' Notes on the Bible

For it is evident that our Lord sprang out of Judah - It is well known: it cannot be a matter of dispute. About the fact that the Lord Jesus was of the tribe of Judah, there could be no doubt; compare Matthew 1:3. But probably the apostle means here to refer to more than that simple fact. It was a doctrine of the Old Testament, and was admitted by the Jews, that the Messiah was to be of that tribe; see Genesis 49:10; Isaiah 11:1; Micah 5:2; Matthew 2:6. This was an additional consideration to show that there was to be a change of some kind in the office of the priesthood, since it was declared (Psalm 110) that the Messiah was to be a priest. The fact that the Messiah is to be of the tribe of Judah is still admitted by the Jews. As their distinction of tribes now, however, is broken up, and as it is impossible for them to tell who belongs to the tribe of Judah, it is held by them that when he comes this will be made known by miracle.

Of which tribe Moses spake nothing concerning priesthood - That is, in the Mosaic laws respecting the office of priest, this tribe is not mentioned. All the arrangements pertain to the tribe of Levi. (source)

Now here is Logic

The Law of Motion states the opposite and equal reaction. However the other Law of Motion states an object in motion remains in motion unless acted on by an outside force.

The governmental structure based off the one law is set free when applying the other law. Not meaning that the first law no longer exists. Indeed the opposite and equal reaction will still occur to the flesh. Yet the opposite of forgiveness is forgiveness. It is like passing through the air. Because there is no resistance, the opposite and equal reaction does not take place. Then the body remains in motion for all of eternity never dying. For this is the Law of Existence.

Therefore it is first for the people to understand the opposite and equal reaction, then they can understand the object remaining in motion. And the bible is how you teach farmers with a second grade education.

Jesus was a rabbi who was adept at the law. The unique thing about Jesus is that he frequently and routinely used the rules of the talmud in order to spar with the pharisees. The way Jesus typically "won" these sparring matches was instead of falling victim to the legalism of the law, he would examine the moral behind the law.

In order to avoid the same traps, we too will need to examine the moral behind the law too. Before we begin to address the authority for offering the sacrifice of Christ's blood and how Jesus was not a high priest in the order of Aaron, we should take a step back and look at the Law's prohibitions against the sacrifice itself. First, in Leviticus, it was forbidden for anyone to consume the blood of any living thing. How then, could Jesus offer his blood to his Disciples, saying "Drink from [this cup,] all of you, for this is my blood"? Seccondly, there were prohibitions in the law against human sacrifice, so how could Jesus offer himself as a sacrifice?

The answers comes from the explanation of these prohibitions. For example, Leviticus 17:11 notes

For the life of every living thing is in the blood. So I myself have assigned it to you on the altar to make atonement for your lives, for the blood makes atonement by means of the life.

Since Jesus is described as the giver of life and living water, this uniquely qualifies Jesus as the only allowable source of drinkable blood. This is because the prohibition on drinking blood was to remind us that we should only be seeking life through God.

Likewise, the reason that God prohibits human sacrifice is that he is seeking to restore us from our fallen broken state. Substitutionary atonement is God's attempt to save us from death. Therefore by sacrificing those which God seeks to save from death, it defeats his purpose. On the other hand, offering Jesus as the sacrifice does nothing but serve God's purpose.

As we can see, keeping righteousness under the law using a wide array of technicalities in order to keep the spirit of the law is an art that Jesus was adept in. This then brings us the the heart of your question. Under the law, Jesus could not officiate his own sacrifice because he was not a Levite Priest. This is why Paul goes to great lengths to explain that Jesus was a priest in the Order of Melchizedek.

In order to understand why this important, we must first understand who Melchizedek was. In Genesis 14, the Bible notes that Melchizedek, King of Salem was a pries of the most high God. What you notice from this passage is that he is making offerings on behalf of Abram. What this means is that there was a priesthood which existed before the 12 tribes of Israel and therefore the tribe of Levi did. Melchizedek is often noted as being the Kind-priest. Paul is trying to make the point that because Jesus is the King of Kings, then he is qualified as a priest in the order of Melchizedek, since he is King of even Melchizedek the king-priest. As such, for yet another reason (according to Paul,) this Makes Jesus not just a priest, but the High Priest because he leads even other priests just as he is King of other kings. And oh, by the way, since Melchizedek wasn't a Levite, Jesus needn't be either. In fact, this qualifies him even more, because he is a priest in the original order of priests , an older order which predate even the Aaronic Priesthood.

(Interestingly, several note that Melchizedek was the basis of the Samaratin's claim to priesthood Authority as well)

And so we see that Jesus is yet again using his expert knowledge of the law and all of its' loopholes in order to save us from our sins.