As a non-native speaker I felt my addition to what had already been said and written was needed. I also thought its premise was clear enough in saying that a word like invite depends on the context it's being used.
Fair enough invite is a verb, but learners shouldn't be surprised if they happen to hear or read "Thanks for your invite!".
As a teacher I think it's helpful to broaden learners views on different usages of the same word. That doesn't necessarily mean encouraging them to use ungrammatical expressions.
If anything I'd be delighted if a student of mine wrote on a quick note to a friend "Thanks for your invite!" and "Thank you for your invitation" in a thank-you card to their friend's mum. That would mean not only were they smart enough to know the difference between a verb and a noun, but they brought it even to a higher level: they understood various pragmatic usages of the same word.
p.s. BTW I looked up ain't on Dictionary.com and its usage is regarded as non-standard, which is very different to informal.