Dave,
Flatbows are pretty safe as long as the back has straight grain. No backing is really needed, but as the blackmockingbird said, rawhide isn't a bad idea.
Personally instead of buying 1 board, 1 piece of fiberglass and one pint of resin to apply it, I'd probably just buy 3 boards.
Red oak is actually pretty strong in tension, it can back other woods and works just fine unbacked. Just look for straight grain and no pin knots on the back.