I know he’s believed to be a theophany by some (I recall that St. Ambrose believed he was), but it is difficult to know. It’s interesting that Christ had an “end of life” when He died on the cross, has a Father in heaven, a mother, Mary, and a genealogy that is explicitly spelled out in scripture. So oddly enough, Melchizidek seems to fit the descriptions of Hebrews 7:3 much better than Christ does if taken in the strictest sense. I imagine that the author meant to apply those descriptions loosely to Christ’s eternal Being as God, rather than strictly to His time here on earth.
Many people say that the writer didn’t mean to convey that Melchizedek was never born, but rather that his life and death weren’t recorded. They would say that his lack of a recorded beginning or end, not a lack of an actual beginning or end, typifies Christ’s eternity. Especially important here is the fact that he had no recorded genealogy. This distinguished him from Levitical priests who inherited the priesthood. I think from this, we can infer that one point that the writer is making is that, “Christ’s priesthood, like that of Melchizedek, was not dependent on the status of his forefathers, therefore, it doesn’t matter that he is not a Levite”. I think this makes some sense in light of the context of chapter 7.
I guess the biggest difficulty I have with the theophany theory is that the writer never just comes out and says it. The writer says that Melchizedek is “like” or “resembles” the Son of God, not that he is the Son of God. Likewise, His being a priest in “the order of Melchizedek” strikes me as being an awkward statement if he was actually Melchizedek. It seems to me that the writer would rather write that Christ was a priest “of His own nature” or of His “own accord” or something less redundant and to the point. As it is written, it sounds as if Melchizedek and Christ are two different people.
God often uses earthly realities to teach about heavenly truths. I think this might be a case of God establishing the priestly order of Melchizedek so that we could look at Melchizedek and later David and Solomon (who were also priests in the order of Melchizedek) and learn something about Christ’s divine priesthood.
But I could certainly be wrong. Let’s say that he was a theophany. Even then, if Christ offered Himself in Genesis, it would break the general rule that the Old Testament contains the shadow that points to the New Testament realities. Obviously this doesn’t have to be an ironclad rule, but it is worth pointing out. As you pointed out there’s no difficulty in Him offering Himself in the Eucharist before His death on the cross, as He is not ruled by time. Another point to make is that His death on the cross worked, in some way, retroactively to save the Old Testament saints. This also shows that Christ’s work on the cross is not constrained by time.
Anyway, I guess that this was a really long way of me saying “I doubt it, but I’m not sure that we could know for sure.”