Solving an issue like this without some measurement is pretty hit and miss, more miss than hit in this case. I dragged out a dehumidifier, which for our purposes is pretty much the same thing as the OP's air conditioner. I took a screenshot of the startup and while the iniitial surge was >4X the running current, the surge only lasted about 8-10 cycles. The compressor starts up pretty quickly. This unit seems to use a zero-crossing switch to control the compressor, as the clean startup here is consistent, not just a random stroke of luck. If it didn't have that, there would likely be an even larger inrush spike at times. In order to really analyze the issue and come up with the proper fix, the OP would have to take some similar measurments to see if it is a very short inrush or a multi-cycle surge that is the problem. However, my guess would be that the OP's AC unit draws about 8A normal running and if it were proportional to mine, the startup surge would be 32A or 3840VA but probably a 0.7 or so power factor. Likely the inverter just can't hack it when presented with that much of a load when it is already on.