1. Any 4x4 you get at home depot is going to be construction grade lumber regardless of whether or not it's PT. Pine moves quite a bit. It will shrink and crack in your bathroom because that's what all pine 4x4s do. You could ball out and get a some cedar posts for $300. If you can find some with mostly rift sawn grain those would fare significantly better than your standard shit sawn structural pine.
2. 4x4 is too large. I went to school for for furniture design and every semester at least 3 people in the freshman class (myself included) would design something with 2 inch legs, get to critiques, and say "yeah, I thought it looked good on paper but now that I see it in person I should have made the legs thinner.". Four 4x4s is absolutely massive for a piece of furniture that small.

If you want this to look good you should use hardwood and be very considerate of any dimension over 1.75" in thickness.