Easy solution for that problem… just fill the bath with Mercury (the liquid metal, not the planet)
Yah, but don’t push the moon down to completely submerge it - again, that gives you volume, not mass. Just let it float and weigh the displaced mercury (easiest if you have a nice big eureka can).
You could also use the planet, melted down. It’s denser than the Moon, so it would work just as well.