The older doesn’t like or respect the younger’s girlfriend and refuses to come to any family function where she is present. He has put me in a position to have to choose whom to invite for what. I think it is an immature and petty position to take on his part but am torn about how to handle it.
Now, if the younger’s girlfriend harmed your older son or his family, then that does change things, so ask him if that’s the case.
Otherwise it’s time to state (apparently, and I think rightly) your principle: Family means accepting you won’t always like who everyone brings into the fold. Explain that you can’t stop him from boycotting – and are saddened by his family’s absence – but, unless there’s more to the story, you won’t enable such a divisive choice.
I’m all for not making waves, but there’s also a point when faking politeness is so transparent that it doesn’t seem worth it.
Should I avoid her altogether at the party, talk with her and fill the time with platitudes, or tell her how I really feel and that we should talk about it another time?
You didn’t give me the one option I was looking for: Say hello and keep on walking.
She might approach you, of course, but for that you can have a clear but courteous response ready, along the lines of, “Please excuse me; I’d rather not talk.” Walk away.