Who knows... In fact, it is likely that even the telescope manufacturer can't advise you here.
If the telescope relies on the AC mains frequency for sidereal tracking then your battery + inverter system should work fine, as the frequency will be divided down from a crystal oscillator and so will have almost negligible drift/error from 60Hz (or 50Hz, for that matter). The "modified sine" waveform is really just a square wave with a peak value the same as the AC mains (e.g. - ~160-170V for 120VAC RMS) but a duty cycle of less than 50% so it has the same RMS value. Some power supplies - and certain kinds of AC motors* - really dislike this bit of chicanery, but there's no good way to predict that unless you have considerably more detail than the pictures you posted.
* - in particular, permanent split capacitor, induction and hysteresis motors really need to be supplied by a sinusoidal voltage