Unfortunately I highly doubt UA will pay for your over-limit fee. I think your best bet is to try to reason with your CC company, point out never late, first time, etc. Escalate it to a supervisor if necessary. And send in a payment right now to get below your limit. I wouldn't count on UA refunding the money quickly.

Disputing the charge is a good idea too. The only problem with that is you haven't taken the flight yet so that could screw things up. You cannot partially dispute a charge, but it may get the charge temporarily taken off your balance while your bank investigates.