Howdy Folks, I have two suggestions. I've tried both and can say they work for using narrow bamboo strips.
1) There is no reason why you can't use two or more parallel strips to back a bow. That way you can use much thinner and narrower strips. It may look funny, but the biggest objection may be that the back is then almost all "power fibers" and may over power the belly. That is easily taken care of by trapping the back.
2) Make the bow from a dense wood like Ipe. Seven eighths to 1 " wide ipe bows are quite do-able and 3-4" bamboo will be plenty wide enough to back a 1" bow.
Ron