Dear oh dear ... I wonder if anyone has ever been sued on AB for giving out such bad information resulting in damage to personal property.
There are some snippets on this thread, and when you add them together, you have got the answer to your question .... "how to remove a stubborn screw"
To use an impact driver you need a certain amount of resistance or something to hammer against and I'm afraid neither a porcelain basin nor a fibreglass one are known for being something you should hammer against.
Having done 100's and 100's of them the way I would tackle it is as CD stated .. give the screw a bit of a turn as though you are trying to tighten it.
I know for a fact that the screws CD is using will take that extra turn without snapping. But I will bet my shirt on it, that the one you are dealing with is made of brass. Or if it's a really cheap tap then it may be chromed steel.
Assuming the soft brass screw snaps, which often happens... you are left with the tap head/wheel spinning freely. Remove it and it is a fairly simple job from then on. Turn the water off ... you can grip the tap body and remove the head assembly. Buy a replacement head set and when you fit the new heads put some vaseline around all screw threads. I couldn't be bothered with drilling the screw out, as it is far easier to buy a replacement head.
So don't worry if you have already damaged the screw head struggling to remove it. Either snap it or as carrust says "drill it out"
... Really surprised no one has mentioned trying ... TNT ...