Thoughts on frustrating situation.

To be clear, if it was not sold at a Chrysler dealer, I'm shocked that they'd even consider paying for a new Chrysler trans.

That's an aftermarket warranty and you need to call them, explain the situation, see what they'll pay for, and as mentioned, study up on your states' lemon laws.

The fact that you haven't mentioned the warranty company or the warranty contract you should have been given makes me think the used car flophouse just took your money and didn't actually buy a warranty. All repairs would need to be run through the warranty company for approval.