We repaired several cracks in the plastic surround by drilling out the end of the crack, cleaning it out and using a flexible epoxy on the face (that was later sanded smooth). The back side of the crack we used fiberglass mesh and the same epoxy, but did sand the area to rough it up before applying.
That was in July, so far plenty of freeze/thaw cycles and sitting on it, standing on it, and cracks haven't reappeared.
We then coated the whole area with Tile-Doc from Sherwin Williams with a HVLP gun including the end cap and medicine cabinet etc. It too has held up very well. The Tile-doc is expensive, we used 2 1/2 kits of the epoxy bathroom paint, at about 70 bucks a kit, but it is a very strong, nearly unscratchable, flexible coating that sticks to EVERYTHING except caulking (which is quite a project to remove every trace of 40 years worth of it.
po armed with a caulk gun - help
shows what we did. Painted all the plastic trim pieces with the Krylon plastic paint and it too has worked well. Just like most projects its all about prep work. I personally wouldn't get rid of the interior parts for your bathroom, refinishing, especially if already removed (we did ours in place) would be quite easy.