It's hard to tell from the pic if the floor is OK or rotten. Maybe stab the floor with a screwdriver along the tub to see if it can poke a hole.
Now that I better understand the risk of someone complaining about sound, I would go with a piece of the thickest, most expensive sheet vinyl you like - not vinyl tiles. With sheet vinyl you have fewer gaps for water infiltration compared to vinyl tile, and fewer opportunities for peeling, gapping, etc.
To lay sheet vinyl, you'll take very careful and precise measurements of the room (considering it might not be square) and buy a square or rectangle bigger than that. Then, carefully cut out the shape of the room, leaving an extra 1 inch or cm on each side. Then carefully fit the piece into the room so that the excess rolls up the walls evenly all around. Then roll it back and apply contact cement under half, then mash it down flat. Repeat for the other half. Then, use a utility knife to cut off the excess that rolls up the wall. Then, use caulk to waterproof the edges and nail in your quarter round over the caulk.
Notes:
The toilet must be removed. Install a new wax ring under it.
The floor must be absolutely flat or every little speck will show through - no staples, little holes, dirt chunks, etc.
Great news: sheet vinyl is cheap enough you can afford to screw up a few times and start over.
Most professionals cut the vinyl to be very flush against the tub, take extra steps to glue down that edge, and caulk it for waterproofing, but do not put quarter round over that section. If your tub makes a straight line, and your cut doesn't work out, you may choose to quarter round it anyway.
YouTube is your friend. It helps to visualize the technique.