Just two quick thoughts, and not sure how useful either is.
First, have you tried different bowls? You mention two different bottles but what about giving him a different bowl? Maybe a wider shallower one, … or the reverse depending on what he was using originally? Something that let’s him tip his head forward while drinking may decrease the amount of water dripping along his chin.
And while this doesn’t address the problem perhaps you could ask the vet is there something you could apply to the dry chin that would help coat/protect the skin from wetness? I really don’t know anything about this in rabbits but if we were talking about a person I’d suggest a thin layer of something like vaseline (not a product I’m usually fond of, … in general I avoid it like the plague but it does repel water very effectively), or a lanolin based cream. As I said I have no idea what might work best on the rabbit, I’m just giving you those two examples to be clear what it is I’m suggesting. Something with a little beeswax might also work, just to help the skin repel the water.
I’m really sorry you and your rabbit are having such a rough time and I do hope you are able to get to the bottom of this issue and find a permanent solution very soon! Best of luck!