If they can get up through the floor, guaranteed they'll have been nesting and playing in the underbelly space and making a mess of the insulation or damaging wiring and other things. Maybe the underbelly material should be pulled down for inspection.
Trying to reach under a shower can be difficult depending on if you have a walk-in shower or tub/shower combo and how large the access opening is. I sealed up all of the holes in our floor (and there were lots). Worst was the duct from the furnace went down into the underbelly and where they hacked a hole about 8x8". I used a piece of plywood to patch that one and foam elsewhere except under the shower I had to use Tuck tape due to restricted access.
A piece of sheet metal is a good idea or a piece of plywood (1/4" - 1/2") should also work. Could glue either down with PL400 construction adhesive. If the hole needs to be an oblong/oval shape, you could cut the metal or plywood as close as you can against the P-trap and then use spray foam to finish it off. If you have enough clearance, you might be able to use a stubby screwdriver to screw the patch in place.
Remember, you should be able to get the P-trap out some day. Maybe just use a few small dolops of glue so you can use a chisel or screwdriver to get the patch off if ever needed. If you have enough clearance, you might be able to use a stubby screwdriver to screw the patch in place.
Mice can squeeze through a hole as small as the size of a dime so you want to seal it well. I'd also seal up any other holes you can find in the floor.