1. Lose the carpet.
2. Get down to the subfloor.
3. Put in wired radiant heating (example:
https://www.familyhandyman.com/floor/how-to-install-in-floor-heat/view-all/
4. Use beautiful ceramic tile to finish the floor.
5. Have a nice throw rug of choice! I use a hexagonal 100% wool rug.
You will NEVER regret it and you will love it everytime you you step onto that floor -- the lovely heat like this is the very finest when doing anything in the bathroom. This is totally doing it the right way. Our radiant heating runs over by the toilet and I have been in places where it is under the tile in the shower -- so nice! Good luck.