“I’ve got” seems particularly strange to me because without the contraction Americans would still just say “I have.” (There are some circumstances where they’ll say “I have got” without a contraction, but it’s mainly when they’re drawing a contrast with what they “haven’t got.” E.g., “No, I don’t have a baseball… oh, but I have got a lacrosse ball, will that work?”)
I think the rule is probably closer to “you don’t contract a stressed verb,” but that’s not terribly useful since there are so few rules about stress patterns. Verbs at the end of sentences are typically stressed, though, so you’re right that ending with that kind of contraction is going to sound wrong to most people.















Mitchell and Webb have a bit about this. Mitchell’s character gets annoyed that Webb’s character keeps talking about how “we beat you in the playoffs.” He eventually asks “Hey, do you remember that time WE defeated the nazis and recovered the Ark of the Covenant? That’s right, you see, I enjoyed watching the film Raiders of the Lost Ark, and so now I have decided that I was in it, and deserve credit for participation in the events of the story.”
(Not exact quotes, I’m paraphrasing from memory)