broken screws on tub pl

Hi, my husband and I installed a bathtup/shower in our basement. All went well until we tried to put the finishing touches on and screw on the plate over the water handle. It had to go through some heavy tile pieces and the screw broke off. My husband is at a loss on what to do about this. He is afraid that it will just break off again if he swivels it and tries again.

Any suggestions?

Is the screw going into a threaded hole or something else? Could you have gotten some thinset in the hole? You might have to drill it out and then use a tap to clean up the threads.