Birch has come up in discussion on another post, and I'll state here as well, that I did not have good luck with paper birch in central Alaska. Hermitking found it weak in tension, and I found it weak in compression. The dimensions you stated might work out for you, because if it is to work, it needs to be wide, thin, and long. I tried pretty standard 2" wide 68" long overbuilt flatbows, and they all failed in compression on the tillering tree. Good luck, heck, I understand the Mojave natives used willow in a deflexed design, and it worked for them. Having had the good fortune to live there as well (the Army only buys the most prized real estate), that is about the only thing that grows straight there, so they had little choice.