an Inverter is simply there to allow you to power any 120v outlets(including the 120v fridge), when you are not hooked up to shore power/generator power.
You can leave the inverter ON even when you DO have shore power, as it's function then is simply to 'pass thru' shore power to all the 120v outlets, and if you LOSE shore power, you still have 120v power to all the outlets.
an Inverter will not run your Air Conditioner regardless, as it is not designed for that, only for typically 'plug in' outlet items, such as fridge, microwave, computers, device chargers, TVs, satellite receivers, etc.
You may find that on your unit the Inverter is only powering the residential fridge, if you have one, otherwise you will need to check each outlet with a fan or something to see if it is powered while on the Inverter...
Most RVs that come with an Inverter are simply due to the need to power a 120v residential Fridge either while you are driving/traveling, or when you have no shore or generator power.
'14 Thor Palazzo 33.3 compact diesel
KingTailgater2 Dish, 100w/5amps roof SOLAR, BlueOx, 100,000+mi since '14... US, Alaska, and Canada... from GA, then NC, now N AL : )