Would be good if you could post some specs on the solar generator unit. I don't know how familiar anyone here is with those. I've never seen one. Having said that, I doubt that it puts out enough power to run your A/C for more than a few minutes, even in full sun. Which I assume is your goal, since you mentioned summer heat. I'm just thinking that "portable self-contained" does not have much solar capacity. Need specs.
Here are the components of my solar generator:
1) TM-2025-RV Battery Monitor Package - pkg includes 35ft of wire and 500amp shunt
2)Morningstar ts mppt - 45amp mpt charge controller
3)Caster Wheels - on bottom of generator so it can be moved
4) Batteries - 4ea 250 ah full river 6 volt
5) Samlex PST-2000-12 Pure Sine Inverter
6) IOTA Battery Charger - 12V 55A - If I need to use a generator this charger should charge the batteries more quickly.
7) solar panels - 2ea - 270 watt monocrystalline from Solarworld
12VDC can come from multiple sources, and as long as polarity matches up, nothing blows up. But what you describe would have the batteries trying to charge themselves, due to the onboard converter. If you turned that off, I think you would be okay. But need to make sure about voltage matching up. Because 4 x 6V batteries in your solar generator may be putting 24V into the inverter. Lots of ways to screw this up. As far as monitoring, if everything is connected properly at 12V, your monitor doesn't know that there are 6 batteries instead of only the 4. Could get more complicated than that if the monitor is a smart one, including a shunt, like a Trimetric setup. But it is probably just a voltmeter.
Really don't have enough info here to give good advice.