I obsessed about this for a long time and finally decided I would just go ahead with the bamboo. As several of you pointed out the fingerboard will contribute a lot of stiffness, plus the neck is extremely thick anyway, so the bamboo's low stiffness is probably not a big problem. Then I started cutting into the bamboo planks and they started bending in all directions. I had noticed this earlier when I cut the pieces for the bending test and hoped it was just that one plank, but no, they all want to bend once you start cutting them.
In glorious hindsight, this shouldn't be surprising. If I had to guess, the bamboo strips are none too flat as they are cut from the stalk, then the flooring manufacturer just trims them square and forces them flat when gluing up planks that are close to the final dimensions. Then there's only a little material removed when they clean up the surfaces and machine the tongues and grooves, so it doesn't have much reason to move. Until I come along and start sawing it apart again, then it all goes wild.
So, this neck will now have a maple spine down the centre to add some strength and stability so it hopefully won't end up shaped like a ski. Speaking of skis, it was so cold in my shop this morning that my coffee started to freeze.
I don't have a piece of maple as thick as I'd like, so step one will be gluing two pieces together with a piece of veneer between them for contrast/disguising the glue line. But I also don't have a long enough piece of veneer, so step zero is stretching the veneer to fit.
Don't know why I didn't cut it corner-to-corner, but that should still be OK. I wanted to join the two pieces edge-to-edge to prevent a gap from opening when I do the glue-up, so then how do you joint the edges of something that thin? There's probably some more elegant way, but I just clamped them between a couple pieces of scrap and had at it with a plane.
That worked better than it had any right to. Then I taped the pieces together edge-to-edge, glued, and clamped. (Back in the house, where neither the glue nor my fingers freeze.) I know you guys like glue and clamp pictures, so here you go.
That came out pretty nice for a trick I've never tried before. The joint will be nearly invisible when sanded. Not that it matters considering it will be buried inside the neck, but it's the principle of the thing and practice for some future project where I might need to do that somewhere visible.
Nearly a month into the build-off and I've successfully glued two pieces of veneer together. Just screaming along!

It's been so long since I've done any woodworking, I'm spending most of my shop time finding things, cleaning things, putting things away, removing rust, sharpening, troubleshooting things that don't work right, replacing broken things, etc. Hopefully this will pick up speed soon.