As others have said, inverters can handle some inrush current but maybe not anywhere near enough to handle a motor starting that would momentarily cause lights to dim if it started on the A.C. line. If you need to measure the inrush current safely and you have some experience with circuits you can make a current transformer from a line inductor salvaged from an old switching mode power supply. The photo below shows how to do it. You may have to experiment a little to get the turns ratio right to give you enough voltage output to measure on a scope but a few turns of insulated wire wound as the primary would probably work. The original winding will be used as the secondary and MUST be terminated by a fairly low value resistor, 10 to 100 ohms would be a good starting point. Without a load (burden) the output voltage could be high.
Wire the primary into a circuit with a incandescent bulb and an A.C. multimeter in series, set to the proper current range, you can view the output on a scope and calibrate the circuit against the DMM for steady state current by adjusting the burden resistor/scope attenuation. Once you have it correct try connecting a motor load and trigger off the current transformer signal and adjust the timebase to display the inrush to steady state waveform on the scope. A meter will probably not react fast enough to display what you are looking for, even if it can capture max readings. A scope will tell you so much more. Either way, for safety, use a current transformer for isolation.
I didn't just try this but I have done a similar thing in the past so it should work.