I used a stave for the belly on my BBO, so I may not be much help on the osage board. I would say as long as the grain doesn't run off to one side or the other you shouldn't have any problems.
The boo side won't matter cause it's glued, but if the grain runs off the side it could split there and go right through the boo backing? Just guessing on what would happen.
Roy from PA builds BBO's using osage boards for the belly and he just finished up a real nice one. I'm sure he can enlighten us on what to look for and what not to use.