At my second battery-change at the zwift-click two screws broke. So I tried to turn out the rest of the screws an they broke again and it was not possible, to get out the rest without damaging the click. The result is, i had to order an new Zwift-Cog. I think, the quality of this screws is really bad.
Do you have photos? It might be possible to still remove them if we can see what has happened: and replace them with screws from a different supplier.
I had a problem with a screw holding in place a Bluetooth antenna in a 2010 Mac Pro. In Apple typical fashion the screw was tightened to EXTREME level. The recommended jewellers screwdriver wouldn’t shift it and just ended up rounding the head. I had to use some pliers to loosen it.