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To add on to Ted's informative post - I have had inverters in class A coaches for 22 years. IMHO you need a minimum battery bank size of 100 amp-h (usable) to power each 1000 watt of inverter power. Using that thought, the minimum power for a 4000 watt inverter would be 8 group 31 lead-acid batteries or 4 LFP batteries of the same size.
A typical 120 volt A/C will require 1 amp @ 12 volts to produce 100 btu of cooling; so 100 amps @ 12 volts will produce 10,000 btu of cooling. That is running current,; figure a surge 160 amps to start the 10,000 btu A/C. Then there is the inverter efficiency to consider.
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Jim & Roy Davis
2016 Hurricane 31S
1961 Rampside in tow
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