Hi,
A 13500 btu air conditioner may draw about 1300 watts. Start up surge would be double that, so a 2500 watt pure sine wave inverter may work. 3000 watts would be better. Go Power and Samlex make units that size.
Beef up the charging path from the engine alternator to the house battery bank with thicker wire. The isolation solenoid may also fail.
Expect to shorten the life of the alternator and when it fails replace it with a larger capacity unit.
Look into after market add on "engine" air conditioners. It may be less expensive than doing the above.