No it won't be true mahogany. There are now a few 'imposters'! All the true mahogany got pretty much used up. Chances are what you have is sapelle. It will still work for your riser so no problems there. However it is a very poor wood for the belly of your bow - it will chrysal before full draw is reached almost certainly. It doesn't have great elasticity.
Bamboo is also very stiff (resistant to tension), this compounds any problems with sub par bellywood.
As for how to use the bamboo for the back of your bow. Use one piece (as wide as you can get), crowned and don't worry about the edges being thin, down the center of the back.
If you have no other options for bellywood other than boo or sapelle then I would go with another piece of boo. Boo is very good in compression IF you heat treat it to a good deep,dark brown.
So my stack would be a boo back, wide as possible, 1 1/2 wide, tapered sapelle core, heat tempered boo belly.
I would make the limbs a pyramid width taper so that you would only need to do a little thickness tapering on your core.
Making the bow like I am describing is tricky because you cannot remove 'bellywood' as you would normally to tiller.....but it will give you a durable bow in the end (even if it takes a few attempts to get the thickness/tapers correct).