OK, I've read over Mage Hand and Mage Hand Legerdemain, and slept on this. One of my campaigns has an Arcane Trickster so it seemed worth putting the effort in.
My conclusion: the AT can pour out a vial (or similar) as a bonus action. They can pour it out while it's above a monster. There is a risk to the monster of it getting liquid on it, but the hand cannot make an attack roll, and the Arcane Trickster can't make an attack through the hand.
Therefore my ruling is that the monster should have to make a DEX save to avoid the liquid. I'd say DEX DC 5 if Tiny, DC 10 if Small-Medium, DC 15 if Large, DC 20 if Huge, DC 25 if Collossal looks about right. On a fail, some of the liquid goes over the monster.
Edit: I absolutely positively would never make picked up/carried items become invisible, though.