Hi, I'm someone who is really into historical bows and also (poorly) makes them. The use of bone in the bending section of bows is a common trope/misunderstanding, it wasn't & can't be used for this purpose.
The best material for the belly of the bow is horn, ideally and most commonly black water buffalo horn. This is the material often mistaken for bone, as despite the name it can also be white. Goat horn may also be used, it's not as good, but I've seen it done before. As goat horns are already in the game, I think this would make the most sense.
Aside from being more accurate to reality, I think this would give goat horns more purpose, and make the players follow historical patterns. Many cultures raised, even imported black water buffalo for the horn.