The LDCI or air conditioner microcontroller may be checking for proper grounding. You should have exactly one neutral-ground bond. When on shore power, that bond is on the shore-side so the RV should have no bond. When on inverter, that must come from inverter, and be removed when on shore.
Other than that, I suspect the inverter is much too small for the cheap A/C's startup surge, which the Kill-a-Watt is dramatically under-reporting.
Keep in mind a residential grade air conditioner will be quickly destroyed by the vibration of a moving vehicle.
Hacking out the LDCI may not be practicable. We have seen reports of the LDCI communicating with the onboard controller on the A/C, to the effect that the controller will refuse to start if it is unable to detect the LDCI. Emulating this may be challenging.
Keep in mind also that lead-acid batteries are a "lie". Using anywhere near their full claimed amp-hour capacity damages the battery - common wisdom is to only go down to 70-80% SOC in daily use (i.e. Only use 20-30% of the battery's capacity). Consider your battery to have 20-30AH.
Lithium batteries are also a lie, but a much smaller one :) as 60-70% of the battery's range can be used daily without deterioration or fire. Only allowing you to use part of the range (and giving you more battery to compensate) is how companies like Tesla and Apple get impressive reliability from their batteries.
Your A/C unit will draw 35 amps or so unassisted by the solar panels. So not even 1 hour of runtime. Watch your draws and SoC carefully.
Preventing solar gain is a great deal more valuable than running A/C. White paint, insulation, whole nine yards.
The Kill-a-Watt is not accurately reporting startup current of the A/C. It is not fast enough, as the peak startup current comes and goes faster than Kill-a-Watt's sampling rate. Cheap A/C units use simple motors (i.e. Not VFD soft-start drives) and startup current will be "locked rotor amperage" which is typically 5-10x running amperage.
So I suspect the inverter is "browning out" and this is tripping the LDCI.
You can fix that with a much larger inverter (with correspondingly-larger idling and conversion losses) or an air conditioner with a VFD on the motor, with soft-start as the least of its features, which will be a more pricey model. This will do nothing to rectify the vibration problem.
Another option is to go 12/24V A/C packs, but you would probably have to build this out of components intended for marine application, at nosebleed marine prices, but the vibration issue still might getcha.
Lastly, you might try a Thermo King style pack as used for standby power and HVAC on semi's. They can often work multi-mode, off onboard gen, battery or utility. Search for a rebuildable one in a junkyard.