question

Turn on Power Assist so the inverter will make up for any loads that exceed the capability of the generator. Without power assist, there's no way for the Multi to prevent the load from overloading the generator. Even with a soft start, you'll exceed the running current during startup. If the generator overloads, it's voltage will drop or will stop producing power completely.

Set the input current limit to something below what the generator can deliver. Otherwise, the Multi will simply pass the load current back to the generator and overload it.

Do NOT disable the ground relay!! This is a safety thing.

Even with everything set properly it is not uncommon for the Multi to assist when loads are added. This is especially true with Dynamic Current Limit enabled because that slowly increases generator loading when the Multi's AC load increased abruptly.

Don't be too concerned if the Multi rejects the AC input momentarily when the AC load increases. This is probably indicating the generator's voltage dropped below the minimum. The Multi should pick up the AC input again in a few seconds. Running the generator in ECO mode exacerbates dropping the AC input because when the engine RPM is lower, the generator can't deliver the specified power and spooling up the generator takes some time. I run a Honda EU2200 with a Multi Compact and an air conditioner with soft start. It all works perfectly with a brief dropout as the generator spools up out of ECO state.

I would manage your loads such that you never exceed the Multi's AC output specs when running on generator. This is because with Dynamic Current Limit active, the Multi won't immediately pass AC input power to the AC loads and the Multi will need to handle all of the increased AC load momentarily.

Another thing to consider is the power your batteries can produce. During the time when the inverter is assisting an AC load starting up, the demands on the battery increase substantially. If the battery voltage sags or it's BMS trips you would get a low voltage warning, alarm or worst case an inverter shutdown.

The Multi will prioritize powering the AC loads over charging the battery so you don't need to disable the charger. This is true when inverting and also when power assist is active. The Multi also limits AC input current based on the total draw of AC loads and charging the batteries.

Since you can run everything on batteries, try a LOWER input current limit.

Also make sure you have Dynamic Current Limit enabled this will smooth the transition.

Try running your generator with ECO off and see if you have the same issue.

The soft start probably has a ~5 minute restart delay so if you leave it it might try a restart, but the same thing will probably happen.

Your low AC voltage disconnect is pretty low. The soft start may be seeing this as a fault. Try RAISING the low AC voltage limit. This will allow assist to kick in sooner.

Look at the max charging current your batteries will allow and compare that with what the solar and Multi charger can provide total. If it's more than that you risk battery damage or BMS shutdown. This will have nothing to do with your air conditioner startup issue. Consider a GX device (eg Cerbo). With it, the maximum charging current can be set and managed to avoid excessive charge current while at the same time maximizing what you get from solar or the Multi alone.