You don't need to chase a ring unless you want to. Most board bows simply come from boards with few runouts, that is, the grain is straight over the length of the board.
If you wanted to chaes a ring, it would depend on how far into the board the outermost full ring were to run. 1.25" is pretty thick, the limbs would probably only run 1/2" thick at most. If you don't mind gluing on a handle piece it'd certainly be possible. And if you just find a good board with straight grain, 1.25" is probably enough for a handle.