Could you run an AC or heater for hours with the battery inverter setup? How does you battery bank recharge, solar or the vehicle? How big is the battery bank?
Why do you avoid using a generator?
in your jeep TJ you could not practically run AC for hours, the battery bank would be too large to realistically fit in a vehicle that size. you could get a small, desktop size electric 200W-ish electric heater if you wanted, it will still need a battery bank but it's more feasible
P=VI, V=IR, P=I^2*R
So for a 200W heater P=200W
your battery voltage will vary but be about 12V, so V=12V
200W = 12V * I -> I = 200W/12V = 16.7 A
Let's assume you use group 31M AGM or gel batteries, they have about 100 Amp-hours of storage. You can't use 100% of the capacity, since this is a non-starter battery you can discharge it down to about 20% so you can use about 80% of the capacity - about 80 Amp-hours.
80 Amp-hours / 16.7A = 4.8 hours (technically the capacity depends on discharge rate and batteries are rated based on a 20 hr discharge, but for the purposes of these kinds of high level calculations you don't need to worry about that. This also doesn't factor in things like voltage loss or efficiency losses from the inverter if you used an AC heater).
So then using that you just need to decide how many batteries you would need. A 31M battery weighs about 75 lbs and is big, about 13"x7"x9.25". Alternatively you could get something like LiFePO4 lithium batteries, an equivalent 100Ah battery would only weight about 30 lbs but would cost a lot (a battle born 100Ah battery is $949).
The next step is figuring out how to charge, you will need to run your engine (ideally with an upgraded alternator) and will need a dc-dc charging system to recharge your batteries. You could use solar, 100W-200W can easily allow you to continuously run something small like a fridge, but for completely discharging a 100A-h battery you would need a lot of sun and a lot of panels.
The reason I said it's not feasible for AC is that even a small portable AC unit is going to be at least twice the wattage, taking a quick look online climate right has a portable 5,000 btu unit rated at 700W. In something like a tent, the duty cycle (the amount of time it's actually running) will be close to 100% if its hot so lets use 100%
700W/12V = 58.3 A ; 80A-h/58.3 A = 1.3 hour. So if you wanted it to run 8 hours you would need at least 7 batteries, which is about 525 lbs if you are using AGM/gel batteries and it's going to take a lot of work to recharge them. Also, at this level you definitely will need to take into account efficiency losses from the inverter. The inverter will draw a small amount of power the entire time it's running, and when converting DC to AC it will also lose a small amount. It could be as low as 50% efficient or maybe up to 90% depending on the load but that's even more losses.
If you have a sailboat, or a really large vehicle you can build a system to run AC on battery but it's not very practical on something the size of your vehicle.