Your inverter is likely a modified sinewave inverter. These are actually modified square wave output of three discrete voltages. For 240 vac modified sinewave inverter the output is about -305v, 0v, +305v. This yields about 240 vrms.
As to your meter. Unless it specifically states that it is a true rms meter then it is a simple averaging AC voltmeter. The AC is rectified and averaged through an R-C filter. The meter is then calibrated to read the equivalent rms reading assuming the input is a sinewave. When waveform is not a sinewave the AC voltmeter is inaccurate.
A modified sinewave inverter will read low on an averaging AC voltmeter.