I would try a screw extractor, you just might have to use a smaller size.

If that doesn't work and you are at the point of wanting to replace everything, I would instead use some waterproof epoxy putty to just permanently attach your replacement handle. Worst case you have to replace the whole valve anyhow, but if it holds you can have a good 10+ years with it.

Edit: Bonus 3rd idea: The screw hole looks like it might be really deep in your picture, its possible you can just turn the broken screw deeper using a drill bit and have enough room to just screw in the valve normally (possibly with a shorter than standard screw).