No, don't work the surface of the bamboo down to thin it. It needs to be thinned from the inside prior to glue up. The bamboo gets progressively less dense, less powerful, and less resilient from the outside in. In other words, it's stronger nearest the outside, and if you thin it from the outside, it will be useless as a backing and fail when you bend it.
How thick is the bamboo?
Unless the bamboo and/or the oak was thick enough to carefully saw off and reuse, I would just trap it and make the bow.