something doesn't seem right here. I have a magnum 2000 watt inverter and it has no trouble running their microwave at well over 100 amps. I regularly use a coffee maker IU's is 105 amps, and a hot water heater which uses 87 amps .. seems to me your inverter is not producing its rated power
Many motors can draw 4 or 5 times their running current when starting.
If the inverter has enough surge capacity AND If the motor starts UNLOADED from any back pressure and quickly maybe the inverter could handle it.
Once the motor starts the inverter can most likely handle the water maker.
Microwaves, coffee makers, HWT. and such DO NOT have that high inrush current of this motor. Yes, electronics can have some inrush but not like a motor in most cases.
He should try the the inverter UPS mode and the handoff from the generator.
ALso he should seriously consider some of the other things mentioned such as the hard start cap. which in reality is just a larger capacity capacitor, or a soft start module, or finding out if the high pressure kick in can be delayed even just a few seconds. It sounds like it is already delayed, it just needs more of a delay.
Try the cheapest first:
--THe inverter UPS handoff since you already have it.
--Then the hardstart cap.
THe last two will take a bit of research:
-- If the high pressure is already on a delay then it is most likely on a relay of sorts which may be adjustable OR replaceable by one with more of a delay possible. If not that way maybe a cascade if the delay is off the control board. Then an adjustable delay relay could be powered from the board output. Once the delay relay is energized set it for 2-3 second delay then allowing the high pressure to kick in. If this is the route chosen then try adjusting the delay for the minimum needed to reliably start the motor and then kick in the high pressure. It may mean some trials.
-- The last would be, at least to me, the soft start control for the motor. But I half suspect that unless the high pressure kick in is also timed later there could still be a problem.
I don't know since I don't have one of these units or an inverter of any size to operate something like this so these are just my musings..
EHEFFA Is the motor a capacitor start type or not? Copy the nameplate info about voltage, current, HP also and post it. Also post a photo of the motor if possible. It all may help. Without that info we are flying blind to a degree.