With the sequence of events you listed, it is the combination of the charger being on, and the HVAC starting up that trips the breaker. The reverse is not true. So the combination of those two activities is popping the breaker. On another website, an individual had the same problem. HVAC turning on tripped the breaker. Only at new GFCI docks. He suspected that it was the new GFCI’s. After two weeks of going back and forth, he decided to try replacing his 30 amp breaker. It was fixed.
A 30 amp main breaker is relatively inexpensive. I would replace the suspect one, that solved his problem. In fact, I would replace both. Make sure when you do that the power and the inverter is OFF. Turn the battery feed to the inverter off after turning off the battery switch. Disconnect the SP cables.
If you don’t want to take that step first, then amp measurements are necessary. Check the pedestals, check your cables with them hooked to the pedestal but not to the boat, check the inbound feeds in the panel (if you are not very familiar with AC electrical, I would have an electrician look at it. You can hurt or kill yourself) Recreate the conditions and check the amperage at the breaker. If it is within specs but the breaker pops, then it is probably the breaker.
With these kind of problems, best to eliminate variables from the pedestal into the boat, one step at a time.