So I did end up getting a Mr. Beams MB750 Stand Anywhere light (well actually two of them because they came in a pack of two: one will stay home and one will come with me travelling).
It is very well made out of nice materials. It is waterproof and really does stand up solidly.
It does exactly what I wanted. I can put it next to the sink and it pops on as soon as I open the door and stays on while I move around the bathroom. I had to fiddle with the exact angle to get it both to turn on and stay on, but it works fine. It has a nice ambient light sensor so it doesn't turn on when it's not already dark. It has a little switch that can turn it off, so it won't accidentally ignite in transit. With 4 AA batteries it should last a long time and be easy to replace anywhere. The light is both bright and diffuse and is a good level. I probably should have gotten the amber version instead though.
Any gripes? Yes, but small ones. I think 20 seconds without motion is a bit short; 60 seconds or more might have been better. The deactivation switch is very small and requires twisting the bottom cover to get at. A membrane button on the exterior might be more convenient and still not compromise the waterproofing. And it would be nice to have a slightly wider angle of acceptance cone, but this one is fine too.
All in all, I'm very pleased with my purchase.