As mentioned in comments, typical inverter switch times are fine for most devices.
The only way I know of to literally have NO downtime is dual-conversion, where your inverter is supplying the AC power 100% of the time, and your grid power is converted to DC and fed into the battery / DC input of the inverter.
That has built-in inefficiencies (thus, wasted power), but is "zero-switch" time as there's never a switch on the inverted power, you just go from float-charging the batteries from AC & solar to drawing down the batteries/solar.
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.
Best Answer
You have faithfully built out your grounding system and kept it separate from the neutral system, which is better than a lot of people do. It sounds like you've done this work to Code. But what you never did is fit the neutral-ground bond in your main panel.
A house has exactly one neutral-ground bond
And the problem is, your generator already has one of them.
Neutral is not ground. Neutral carries the ordinary return current that the circuit needs. Ground carries only fault current during problems.
However, neutral is tied to ground in exactly one location. That's a very big deal. The point of the N-G bond is to assure that your electrical system isn't floating at thousands of volts compared to actual earth. Because the insulation in appliances isn't rated for thousands of volts.
But if you have two of them, that is very bad. The ordinary current flowing on neutral will then "split" at one of the bonds, and some of it will travel over the ground wires to the other bond. That means safety ground isn't safety ground anymore, it's a working neutral wire. Certain failures can then cause grounds to be energized at mains voltage - which is the last thing you want.
And the neutral-ground bond (1) is always at a grounding rod point. You don't tie ground to earth at junction 1 then have a neutral-ground bond at junction 2.
Right now, your neutral-ground bond only happens through the generator. That means it's not at the same place as the grounding electrode. That's bad.
Put the neutral-ground bond where it belongs - in the main panel where the grounding electrode comes in - but then you'll have two of them when you connect the generator, and that won't do.
The cheap, but at least Code, answer is to remove the neutral-ground bonding in the generator, and then put it back when you unhook the generator to use it as a portable generator (where it needs to ground its outlets).
The cheap, non-Code answer is to remove the ground wire from the NEMA 14 inlet where you plug in your generator.