marine batteries / deep cycle means they are optimized for delivering low current ;levels over along time, as opposed to starting batteries that need to give a large burst of power to turn over an engine.
Batteries are listed in terms of "amp hours"; ie 500 amp hour battery will give 1 amp over 500 hours, or 500x 12v = 6,000 watts.
HOWEVER! the amp hour listing is a little disingenuous, as the power available drops off rapidly as battery draws down...a charged battery at 13.4 volts will hum along; when it gets to 12 volts it does OK, under 12 volts it still has power to give but not much.....if you were running a light, you'd have first bright, then adequate and then a dull glow.
A/Cs as you mentioned have asurge every time they start; if you're running an inverter I'd get at least a 1000 watt job.
As power available drops, your A/C will have to work much harder; I doubt that, even though on paper you have 12 hours capacity in a 500 amp hour battery, that you'd get more than 8 hours out of a system.
You'll need at least a 10 amp charger to keep up with the load.
Many marine and many R/V or truck supply stores sell 12v A/C units.......consider doing that rather than running thru an inverter and losing 15% to resistance in this extra step....you can also look at the draw in 12v on the data plate and know exactly ow much is going out and how much is going in.........