I think that you got what you got there. With the right 1:1 transformer you might knock some of the edges off the waveform. About the time you get it tweaked, it will probably die anyway.

However, I think there is a LOT of misinformation out there on how "bad" cheap inverters are for electronic devices.

I used one for years running computers, televisions, stereos and the like with absolutely no problems. These devices had switching power supplies in them which seem to eat whatever you feed them.

Motors, however, do not like them much and run hot.

Opinions vary on this so I just cite my real world experience running cheap inverters. I now have a sweet 2.5 kw Outback and it runs anything I want it to run within reason. Of course it took a big wet bite out of the budget. Silly expensive hobbies I seem to have.

Cheers,.

TomW

The Truth is the Truth, even if no one believes it; and a lie is a lie even if everyone believes it

It is.  What you need is a filter.

Unfortunately the filter consists of very large value inductors and capacitors, that are capable of handling the sort of currents produced by your invertor.

Other ways to change a square to a sine is with something like a Sola constant voltage transformer. They seem to be able to clean up anything, but there is a cost. They are big, heavy, and noisy. They run hot and use a lot of current to keep the primary coil energized. They're not so bad if you're running at the transformer's maximum load, but efficiency falls off dramatically if you're running less.