Ok, if it was mine and I wanted to repair it, my first thought would
be what is on the other side of the wall? If whatever is there is
easier to repair than the tile (which would be pretty much anything) I
would probably try to open up the back wall and work through there.
That is assuming that I had screwed up the faucet beyond redemption
when I tried to use a dremel tool to remove the old seat. Dremels
should probably be put on the list with duct tape, silicone , sawzalls
and a BFH as fix all requirements.

I'd stick my head in that cabinet first, and see if there's
a way to get at the stub-out from underneath, first.