There is no answer that is correct for every situation. You need to know a few things.
1. If you only charge the batteries, nothing else, yes that adapter is perfect.
2. What size is the circuit that you are plugging into? A residential plug that will accept the adapter you linked to can be either 15 or 20 amp 120v alternating current. A 15amp MIGHT start your air conditioner if you turn off everything else, EVERYTHING!! If it starts it might damage the AC in the long run. If you plug in anything else, it will trip the breaker. There possibly could be damage to the AC if it is run with low voltage for extended time. (the AC manufacturer's specifications say "20 amp" circuit is the minimum size power supply and also limits the length of 12 ga wire in the circuit, if memory serves me, to 24 feet)
2. If that adapter is rated 20 amp AND
your receptacle is rated 20 amp AND
that receptacle is on a 20 amp circuit breaker with the correct ga wire AND
the total length of wiring in that circuit is short enough that there is little loss of current, then I would say it will run the AC unit without damaging anything. BUT
, if you turn on something like your hair dryer it will begin to overload the circuit.
3. Even if you have a properly installed 30 amp RV circuit you still have to consider every item you plug in. Remember the converter is always plugged in, using 120v power to charge batteries and power all 12 volt
items. You may have the refrigerator on 120V. An example: if you run the AC during hot weather or a small electric heater in the winter and at the same time you want to use the electric coffee maker, all is well. But while you are waiting for coffee you decide to turn on your hair drier. Something will trip.
You will learn how to manage loads pretty quickly, because tripping breakers and melting cords will educate you. Replacing an air conditioner gets expensive!
IMO, you will need to have 30 amps if you are living in it full time, or forgo air conditioning and electric space heaters.