I use stainless bowls because I don't like those (due to the cleaning issue and because they can clog without people's knowledge).
You could try a baby bottle brush and hot water. If you boil plastic enough, it will breakdown a bit and can leech into things, so just consider that boiling heat and plastic aren't always the best long-term. Can you reposition the bowl in such a way that it is not getting pooped on? OR can you find a fully stainless water dispenser that would be safer to boil repeatedly?
You don't want to boil much vinegar around birds, as it can be a problem, but a tablespoon or so, when mixed with a decent amount of water is okay...Boiling lots of vinegar can harm them..like, when people pickle things, run it through coffee pots or dishwashers (straight or in higher concentrations)-- this is generally when it becomes a problem.