You see, Barbara, it looks much more homely without a door
yes, that would annoy me a bit too. Why not take a smaller door?!
Perfect!
But then a small utility sink in the laundry room would still be possible, right? I would consider that.
How about a small hatch to the tub, where you can pour the water bucket towards the tub
No, just kidding... if you don't have a load of laundry twice a day, the water bucket shouldn't be a problem.
I would put/mount/fit the two washbasins on one continuous countertop, so that a wooden or work surface goes from the wall to the privacy screen. Underneath, you can then "hide" the towels on a second shelf