To test a PSW inverter you need a good quality 'scope.. but you can test most of it with just meters
If you have TWO Volt meters, one if which states it is a "True RMS" and the other which does not.. Do they agree within a volt or two? MSW inverters will often read different (low usually) on the NON RMS meter.
Also meter shore power with the same two meters to insure they agree.
Frequency can be measured with some multi-meters or a Kill-a-watt but only after the above test shows good.
Total capacity can be measured with the Kill-a-watt and a bunch of 100 watt light bulbs or lamps, or other known value loads in the range.
but for the waveform. ONLY a 'Scope is for sure.