I screwed in too hard... (my mistake!)
The 10 year old shower is in very good shape, looks nice/no leak, etc. So won't need to remodel until a good while later.
As for the other end, it already has rust building up around that new stainless steel screw I just put in two days ago (I did pour in some drain cleaner the other day). This tells me the rust issue from the cast iron pipe might be an endless problem in any case and will cause the screws to stick again later even if I managed to make a new hole this time.
So perhaps a snap-in strainer is a good choice regardless... .