We teach that God is changeless. But at some point the Second Person of the Trinity became man. How is this hypostatic union not a change in God?

Among other things, the answer to your question is caught up in a very deep mystery of God's relationship to time. In fact, many of our questions about God, and our struggle to understand Him, go back to this mystery.
 
The fact is, God does not dwell in time as you and I do. Rather, He dwells in the fullness of time, a mystery we call “eternity.” The word eternity does not simply mean a long, long time. Rather, eternity means “the fullness of time.” That is, past, present and future are all experienced at once.
 
To illustrate eternity, consider a clock. Let us say that the current time is 12 noon. For us who live in serial time, that means that 3 PM is in the future, and 9 AM is in the past. But now move your eyes to the center dot of the round clock face. At that center point, all three times 12 noon 9 AM and 3 PM, are equally present, and have the same relationship with that center dot.
 
This is an analogy of what we mean by eternity. Thus, for God, who is eternal, the future is just as present to him as the past. God is not waiting for things to happen, neither is he reminiscing on things past. All is present to him in a comprehensive “now.”
 
This is mysterious to us but helps illustrate why our questions about God are often wrong, even in the way we ask them. Thus, your question about whether God “changed” when Jesus became incarnate, presupposes that God lives in time as we do. But he does not.
 
While this approach may seem to be avoiding an answer to your question, we do well to meditate on the truth that we do not even know how to word our questions properly, let alone get simple answers. Change, as we define it, presupposes serial or chronological time. But God is not there. God is beyond such categories.
 
Thus we are right to conclude, as you observe, that God does not change. And yet He is able to interact with us who do change. But how exactly this is accomplished is mysterious to us and is caught up in our limited capacity to understand the mystery that we call time. In no sense is God waiting around to do things, like becoming man, nor is He changing as time goes along. All is present to him in a comprehensive now. God simply Is.