For an Inverter to produce enough watts to run an air conditioner you will need to provide 120VAC @ around 20AMPS AC Current.
This is going to require around 12VDC at 200AMPS Current. This is around 2400WATTS.
Then you will want this Inverter to run for many hours... Can you image a battery bank that will produce 200AMPS of DC Current to run for 10-12 hours or so over night. You also have the problem of not discharging your battery bank lower than its 50% charge state or your batteries will not last very long. Batteries are expensive.. So this means you will have to come up with a battery bank that will produce 400AMPS of DC current so you can only draw 200AMPS for the 10-12 hours.
Most batteries specs will tell you how many minutes you can draw 12.0VDC for so many minutes. A Trojan T-1275 Battery says it will will produce 12.0VDC at 75AMPS DC current for just 70 minutes. That's just a tad over one hour. Of course this will totally discharge the battery so you really only get to see 35 minuets of use if you don't want the battery to drop below its 50% charge state.
I would think you are talking about 8 to 10 Trojan T-1275's batteries in parallel to even start thinking about this process...
You will also want to use a HD built Inverter... Not one of the $150 types...
I am no expert here so you need to run these type of questions through the experts on here to get the real skinny of what it takes to have an inverter operate your 13,500 BTU air conditioner.
I run a 600WATT PSW Inverter at half power here alot from 6PM to 10-11PM at night only drawing around 22 AMPS from the 255AH battery Bank and by 8AM the next morning it is right at its 50% charge state (Around 12.0VDC)... Then I have to run my generator for about three hours to charge my 255Ah battery bank back up from its 50% charge state to its 90% charge state so I can do this all over again the next day/night run off my batteries. I can do around 12-14 cycles of this 50% to 90% charge states before I have to run a full 12 hours full charge to keep from hurting my batteries. Its not as easy as one would think haha...
I am a very long way from running a 13,500 BTU Air conditioner doing this for sure...
I don't think you will find this to be a feasible thing to do...
Using 10each Trojan T-1275s 12VDC 150Ah 82-lbs batteries will weigh around 820 lbs - Where in the world would you put these batteries in your Toyota Minnie Warrior camper...
Just me blabbing along here - please check with the Battery experts on here to get real numbers... Ask Pianotuna how many batteries and AHs he runs in his full time live in Motorhome... I'm not sure if Don runs his Air Conditioner off the batteries or not...
Roy Ken