Change Irregular Shower Handles
That's a shame, those are some cool-looking handles..

I know you said it wasn't, but they do look like allen screw holes, just a little corroded. Better hit them with some PB Blaster and let them soak awhile, while you try to find the right allenhead. I predict you're in for a battle with those screws.

Even when the screws are out, they won't give up easily. Those handles are on splined stems sticking out of the wall, and more than likely they are a little corroded as well.

Tap with a hammer, pull, pry, spray PB, tap, pull, pry. Stay vigilant. It will be a long day removing those handles.

Once they are off, splines are easily matched up at most hardware stores. Well, easier than removing those handles, anyway.

Have you tried just unscrewing the porcelin section yet? Maybe you can find something that will just replace that?