I just replace a section of my slide for the exact reason you stated. Mine had 3/4” plywood covered in that plastic wrap like in the link you posted, above that was a layer of insulating foil backed material and then the carpet over that.
This is what I did:
Cut out bad squared off
Cut new piece to fit.
Coat new piece in water seal paint
Coat cut edge of existing flooring with water seal
Wrap new piece in that plastic stuff
Wrap edge of existing floor with that plastic stuff
Put new piece in (two people would have been helpful)
Attach two pieces together using metal straps and screws
Cover the seam where the two pieces butt together with the black plastic stuff
Put insulating layer and carpet back
Clean up
Admire your work and think of just how much money you saved doing it yourself.
It took me 8-10 hours to do mine, I split it in two days to let the water sealer dry.
