Have posted on this subject before. It is quite possible to run a/c from solar and battery suite. Have read posts by Pianotuna, jimindenver, sushidog and RoyB before.I find their posts interesting and useful. Have crossed posts with each.
Power management is required for full-timing and one probably cannot reasonably power an air-conditioning unit 20 hours a day from a combination of solar and battery suite. We have a energy hog Dometic air conditioner and the BMS shows that it is pulling more than 1750 W from the battery suite (there are efficiency losses in the inverter) when it is running. We have had no problems running it for 3.5 to 4 hours in the afternoon and were still at 60% to 70% SOC.
We generally do not hang about inside the rig on hot days: either hiking or generally wandering about or reading in the shade of canopy or other shadow of the rig. We then turn on the air conditioning at dusk for an hour to cool down the main cabin/bedroom and then use 12 V fans to make things quite livable. We have 12 V outlets above the lounge chairs and above the bed.
Several of the respondents mentioned above have gone to 500W window air conditioning units and have found that they can use the air conditioner on sunny days. Got excited about these units but Elaine said we do not need it - just go up in altitude. We are almost always in Rocky Mountain/Sierran West. We generally follow the advice of Hermes and jimindenver: go north or up in elevation.
We have a 8.7 kW-hrs LFP battery suite (at 48 V nominal specs or 9.6 kW-hrs at 54.4 V float) so we have at least 7 kW-hrs usable at 20% SOC. We also have 1420 W of solar and yesterday were getting 1400 W according to TriStar monitor (TriStar MPPT-45). We could add panels to permit running a/c all day but Elaine again said 'we have enough - just go where its cooler!" The battery suite weighs approximately 250 pounds. Lead acid suite with 7 kW-hrs usable would weigh somewhere between 800 and 1000 pounds and that would be more than the frame would care to handle in the front bay.
We chose to boondock/drycamp (drycamping to us is Forest Service/ National Park CGs or mootchdocking in our kids' backyards; they all have 30 amp plugs for us but we have not used them in two years) since we do not like RV parks (we may use an RV park for one night every 2 to 3 weeks to dump tanks and wash clothes - the shame of it all). An RV Park is going to run at least $20/day and 365 days means we save over $6K/year. We amortize the system in 3 years.
Reed and Elaine