There are two issues here: the on-board charger, and the thing that so many people call a charger, really an EVSE, which has a power supply in it to power the electronics.
The latter probably won't care much about a non sine wave, as it's low power and as suggested, the first thing is rectification and smoothing, which actually works better with square waves (the peak current is lower as power flows for more than a tiny fraction of the wave cycle).
But the on-board charger (OBC) is well past the power threshold where it has to be "power factor corrected", in other words, it's expected to look like a resistive load, where the current is in phase with and proportional to the voltage. So the OBC will have a boost stage so that it can draw power even when the input voltage is low. It will be assuming that the input voltage is sinusoidal. How it reacts to a square wave with a significant fraction of the time at zero voltage is unknown. Some will work happily from a DC source. My guess is that it will work OK, but it certainly might not. OBCs are expensive, so I really would not want to risk it.
> The price difference is remarkable,
That used to be the case, but there are some fairly inexpensive pure sine wave inverters these days that are pretty reasonable quality. These days, I would just not consider modified sine wave inverters for any purpose.
For an EV only situation, I'd not bother with split phase, since the OBC doesn't want it, and it basically means duplicating the whole back end of the inverter. Some models require you to buy two 120V inverters that you can then put in series. Even if you find you want to add a little lighting or other loads at a later point, the rest of the world copes pretty well with single 240V sources, so you could always buy an appliance intended for European or Chinese markets and just modify the plug.
2012 Leaf with new battery May 2019. New to me June 2019.