I have used strips of smaller bamboo on the bellies of recurves for a lightweight stiffening material. You can toast it a little and bend it nicely.
Is it thick-walled or pretty thin?
I also have used whole canes to make some lashed on tips for a bow.
2" is big enough to get backings from, for a narrow, stacked design, as mentioned above. You can even choose out the best parts and splice two of them in the handle.