Stryker,
i started building with red oak, and faced the exact same challenge. i spliced two pieces of vertical bamboo, sanded it thin, and backed red oak and when i found ipe at a decking supply i backed that too. it was a waste of time, money, and effort. the best lesson i learned when i started (and learned the hard way) is go ahead and use good materials. it may be more expensive to order, but your chances of success rise dramatically. the frustration level of breaking multiple bows might be avoided.don't just order 1, save the shipping cost by buying 10 and be set for the next bows too, you're going to end up building them anyways.
having said that, there are ways to stay inexpensive...until you order something. find your local lumberyard, i found a local one that would cut hickory strips for me 2 x 1/8th x 72, and if i bought 10 it came out to a few bucks each. go to goodwill, and even though they might look at you funny when you walk out with several cheap ladies silk garments its worth it. try brown paper and TB3 it works. you can also rip a 1/8 strip off your red oak, flip it and glue it back on (add a little relex while you're at it).
You can however use flooring for bellies and lams, its practically actionboo anyways.
good luck, hope it helps, but don't waste your time with the flooring for a backing, just wanted to save you the bloody nose i got