I think we're confused with your question/situation:
do you mean the 'engine for the vehicle' is running?
do you mean the AC for the vehicle, or the AC for the coach?
do you have a 'residential' fridge(120v) that only runs on the Inverter or shore/gen?
if you are running the 'generator', then no, the inverter does nothing but 'pass thru' power to the inverted outlets(like the fridge), and the Charger then charges the house batteries.
if you are only running the vehicle 'engine', then the Inverter will provide 120v power to any outlets it is attached to, such as the fridge. Your vehicle's AC will run, but your coach's AC will not run on the inverter, only on shore or generator.
'14 Palazzo 33.3 bunkhouse 34' diesel
KingTailgater2 Dish HD, 100w/5a SOLAR, BlueOx
100,000+mi since '14 - US, Alaska, Canada - Hiawassee, GA! (also, '14 Gulfstream Amerilite and '07 ForestRiver Rockwood) : )