It's odd the click is coming from the dash area. I would think the front roof AC should be tied into the 110V breaker box only. Maybe someone else has more knowledge of this.
Hopefully your generator didn't go low voltage when it died and fry the AC.
Typically you should let the generator run for a few minutes and stabilize before putting any loads on it.
2009 Georgetown 350TS (bunks)
400W solar, 440AH 6V GC2
2009 Ford Edge AWD Ltd towed
2011 Honda Fit Sport towed