Bamboo is in principle excellent for such parts, as long as you get the straw yellow natural material. This is not brittle at all and cuts like a dream. No surface tear-out and smooth cut. I use it a lot and actually the entire frame and gantry of my CNC machine is made from Bamboo plywood.
Do not use the heat treated "Carbonized" amber or brown variety. This is indeed brittle and much weaker.
That said, bamboo plywood usually comes in 3-ply only. If less than 3/4" thick it has a tendency to warp, making it unsuitable for such precision parts.
As for Baltic Birch, you can get aircraft rated sheets, e.g. from Aircraft Spruce. The 5-ply 1/8" or 12-ply 1/4" should be the ticket for such mechanisms.