Aircon power from 240v Inverter

The problem is a factor called the locked rotor current.
This is the maximum starting current that the motor can draw under load conditions and is normally around 6 times the listed running current shown on the name plate.
So, at 5.4 amps running, (some of that will be the fan/s, which will normally start before the compressor on cooling, and after it on heating) you can expect around 33 amps starting current, and this is on the 240v side, a little less for an inverter type AC.
So your instantaneous power demand at startup on the output of the inverter is about 8 kVa (close enough to 6500 watts at 0.8 power factor).
On the input side, this equates to somewhere around 550 amps.
Now, whilst this is only for a short time on initial startup, and could possibly be handled by a 2000w continuous rating inverter with a large connected input capacity, or a decent generator, once the AC has been running and the compressor turned off on thermostat, the next start will be under load for longer due to compressor head pressures, and so is likely to trip the overloads in the inverter or generator.

Due to the high current draw, severe voltage depression on the 12v side is very likely with a small battery capacity such as yours, and this is likely to cause similar undervoltage on the output side, which will cause the motor to draw its maximum amps and yet still be underpowered for startup.
There is a very real chance of over-driving the inverter electronics with subsequent failure.

Even without the above constraints, your 5.4 amps on the 240v side equates to around 110 amps on the 12v side.
Allowing for Mr Peukert, this will give you around 75 minutes running time till your batteries are at 50% Soc.

You would also be running your 1500w inverter almost flat out, which, although rated continuous, they do not like to do.
Locked rotor current is normally listed on the motor name plate as either a figure in amps or kVa, or a letter corresponding to a fixed list of values.
Your inverter would likely handle the hot water or the 3 way fridge more readily as there are no transient events.

You then have to recharge those batteries, and if doing so with solar, the catch is that the hot day (when you are most likely to use the air conditioner) is less conducive to good solar input than a cooler day.

Strewth Brian01, I wanted details but you type too quick ;-)

The locked rotor current is 20A, the three batteries run off the car and a 150W (kW??) solar panel.

The 110A above would be when it is cooling, not constantly after the van reaches temp?? Not sure what the "head pressure" does to the starting current.

Treat me kindly :-)

The 110 amps is only whilst the compressor is running, but these very small units that they put in vans, coupled with the relatively high heat energy intake of the van itself (thin walls, lack of insulation, lots of windows, lots of body heat for small area, cold air loss through door vent etc.) means that the compressor will have a duty cycle somewhere north of 80%.
So.. if you multiply 110 x 0,8 you get 88 Ah for an hour's running.
That gives you about 2 hours down to 50% Soc, and that's not allowing for any other loads, but it will be less time than that due to both the starting current draws and Peukerts effect.
Increased head pressure, once an air-conditioner has been running makes it harder for the compressor to overcome inertia, and so increases the high load starting time, this can be more prevalent when duty cycling is high due to lack of time for pressures to equalise.
Roof top air conditioners also have a greater propensity to high head pressure due to the generally higher temperatures of their condenser units

Guys, I did a lot of research on this when I last built a TV OB truck intended to run on portable generators when required. A good Inverter air conditioner has a very small in-rush / start up current or almost negligible locked rotor current. Some literally run a large stepper motor and ramp the compressor speed up from dead stop to whatever is needed. Others run a variable frequency drive to a more conventional motor to control speed. We ran a 24kw cooling capacity dual head Mitsu unit off a 6.5KVA Honda inverter generator and never saw north of 3.2KW current draw with no discern-able spikes whatsoever. It was clean enough that I was also able to run some of the equipment power off this gennie rather than fully load our second one. And as for inverter gennies not liking inverter power supplies connected to them... We never had an issue and this thing had about 120 power supplies connected from small to quite large.

I may be corrected here, but to my knowledge RV units in general, and Dometic and Aircommand ones in particular, are not inverter type. I have never seen any of their advertisements stating that point, which I would expect them to do if it were true.
Would be happy to know differently.
The heavy thumping when the compressor starts and stops has been one of the biggest complaints with these units, and still seems to be the case with the newer ones.

Thanks Pat, and John - it's a new Ibis roof mounted so not sure what startup it draws - looks like a gennie might be the way to go for the "odd" occasion I might need it. Small van (17') so would only take 10min to cool - I was assuming a reasonable (say 26deg) setting in a small van would not draw too much.

Yes they are, but LFP batteries do not suffer severe voltage depression under heavy load as do LA types. This is the biggest problem with starting a motor on an inverter.