First thing is to clean up the slot. Second is to get some penetrating oil soaking on it. Third is to get an impact screwdriver that you hit with a hammer. I suspect it will either let loose or break off the head of the screw, either of which should liberate the cover.
That said however, sticking a softener drain tube down it probably violates air gap requirements. You may know that and be planning to make a standpipe to put in it that you can then have a proper air gap on.