Thanks for the responses, guys. Incidentally, the current president of the co-op is actually my dog-sitter. I floated this to her on the dl and she was so outraged she called all the rest of the membership committee, which was not what I wanted at all--I was trying to avoid escalation--but at least all of them thought that my neighbor's demands were laughable. Same comment from two of them: was he drunk? They were going to "take steps" or something but I walked them back from that for now.
They're now insisting that the dog is allowed to play in the backyard, he just can't piss or shit there. I'm not sure how that would work, honestly. Anyway, I'm in a weird position where I have to choose between communicating with the rude husband or his politely passive-aggressive enabler wife. Not sure which is the better option.
Just (politely) let them know you've already spoken to the co-op, you have their backing, and if you agree to keep the dog from doing it's business in the backyard, they should be willing to blaze somewhere else, and finish the painting. If not, let the (co-op) dogs loose.