I have been dealing with the same issues.
I run two different welders offgrid from inverters. One main unit is a 200A AC/DC invertrer tig (that is also a Arc welder) the other is a smaller 140A inverter tig (also a arc welder) I also run a 60A plasma cutter to.
The issue you are having like you say isn't that the load is to large for the inverter. It's the kind of load. When manufactures design there equipment they are counting on most people will be using it powered from the grid. The grid has a huge ability to absorb things that can create problems with battery powered inverters.
Harmonics and all kinds of feedback issues can have your inverter doing all kinds of crazy things. From experience even a tiny adjustable electronics rework heat gun that only has a 250w element can result in a 85kg 6000w continues high quality inverter shuddering and carrying on (making all the lights flickerin the house and fridge make strange noises) It's just the feedback from the device.
No issues on the grid as its just way to massive to have a effect on it.
To often my wife would come up to the workshop and say the fridge is making strange noises. I would look up from the welding table and think.... hmmm all the dials on the front of my welder that are used to chop up the output and make it pulse and tweek the frequancy output in every possible way do result in a great weld but are hell on my inverter and are feeding back and are having a similar effect on the house inverters nice stable clean output.
Another good one was when my wife mentioned she could hear the rhythmic sound of my 3cyl 16A air compressor coming through the vacuum cleaner when she was using it in the house:lol:
So the way I fixed the issue was to install a separate inverter for the workshop. Now the workshop is completely isolated from the house and all my nasty loads from welders and induction motors have zero effect on the house. Also its great not having to think befor I crank up a 4000w welder. I wonder if my wife is vacuuming with say the dishwasher on and a few other loads at 3000 - 4000w in the house. I can just go full tilt and If i trip the inverter it has no impact on the house.
The dedicated workshop inverter is just a inexpensive inverter as I think for it's application top shelf quality isn't needed. It has a 4000w continuous rating and 8000w spike loading. Haven't trip it out yet.
All the gentle clean demestic house loads from appliances run through the selectronic inverter and it has a easy life now.
Offgrid 2008, Selectronic PS1 6kw/48v inverter, x 2 Midnite solar classic 150 MPPT, 3960w NE PV 24 x 165w BP panels, 4200w NW PV 21 x 200w DAQO PV, 16x400ah lithium.