an isolation 120 to 240 transformer, split phase is irrelevant since it is single phase output. But it will probably be cheaper to just buy a split phase inverter.
Looks like that tiwan inverter from pics. I like the idea of transformer separate, and they will ship just boards for repair. But that is single phase input so you can't just swap the board, you will need to add connectors and wiring at least.
I'm guessing since you don't even understand your power grid to answer the simple question, telling you to open the case and see if the AC is passed through, or is fed into the transformer just like the DC volts are would be pointless.
But if you want to test since it says 120v input, use a 1 amp fuse on the hot line, that will limit power to 120w . With no load you should be safe to turn on and check output with vom.