Why must the hypostatic union be viewed as taking place at Jesus' conception? What are the reasons why Trinitarian orthodoxy holds that the Word had to make his hypostatic union with the man Jesus Christ at the point of his conception in Mary's womb rather than later at his baptism by John the Baptist?   Is the view simply that the Word was incapable of uniting with a full-grown man?   

The reason I'm asking is I don't see anything in John 1:14's "the word became flesh" that specifies when that took place.  From a purely scriptural point of view, I don't see that it matters whether it took place at the conception or the baptism, that the word united with Jesus. But I'm sure the framers of Trinitarian orthodoxy must have had some reasons for placing it at the conception.
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:


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.
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.