"You didn't have a coat?" should be correct! "Didn't you've a coat?" isn't even proper English!
The correction that pops up on my screen actually says 'Didn't you've a coat?'
It's a problem with Duolingo's automatic equating of "you have" and "you've". In most cases it's legitimate but obviously here it's not, although the algorithm can't tell the context. Ideally they wouldn't show "you've" as a possible answer when we haven't included it as one, and only allow it as an answer provided by the users, but this isn't under our control, it's under Duolingo's.