I guess the battery is powering the inverter that is running the air conditioner. Right?
I have trouble believing that the battery would last very long under these conditions.
To be sure I would take the battery to someone for a load test; not just testing voltage at the terminals.
Even new batteries can be bad too soon.
Where to buy? Try Batteries Plus.
Dick_B
2003 SunnyBrook 27FKS
2011 3/4 T Chevrolet Suburban
Equal-i-zer Hitch
One wife, two electric bikes (both Currie Tech Path+ models)