Batteries first. You'd be better off with multiple 6 volt golf cart batteries. I highly recommend the Trojan T-105 batteries for inverter use. They are rated at 120 amp hours and you will need to tie two together in series (+ to -) to make one twelve volt bank. To run your A/C, you're going to need at least 4 total if not more. 8 would be ideal but that is a lot of weight to carry around.
Inverter next. Stick with a high quality brand such as Xantrex or Trace (both made by Xantrex BTW along with some other brands.) For type of unit, try to stick with a pure sine-wave form inverter and stay away from modified sine wave or square wave inverters. They do not like to run sensitive or inductive loads very happily. Without knowing the specs for the A/C unit beyond 5.5 amps, I'd recommend at minimum a 2500 watt inverter.
You may also think about going with a 24 volt inverter instead of the more popular 12 volt. This will cut your amperage draw from your batteries in half. You will need to wire your batteries for 24 volt though which is not a big deal since you'll need multiple batteries anyway except that you'll need multiples that will work out to 24 volts.
The battery cables are very important too. Runs should be very heavy cable, minimum of 3/0 gauge - 4/0 is better, and be kept as short as possible. The inverter can be mounted in or out of the engine compartment. Ambient heat will affect output but the higher end units have cooling fans and larger heat sinks and are able to handle it. Mounting location should be determined by where your batteries will be mounted to keep cable runs absolutely as short as possible. Don't mount the batteries in your engine compartment and then run battery cables 25' to the inverter.