driving while the charger is plugged in?

[JoulesThief]

I don't know specifically how picky the e-Golf is about the waveform, but it is well known that some EVs will shutdown charging if they don't get a clean waveform. A clean waveform that is stable in voltage and frequency can only be produced with an inverter generator. So, I don't think the EB series Honda generator linked above is the best choice. The Honda EU series of inverter generators is known to work well with picky EVs.

It was just one of the cheaper Honda 10K generators. I can't find an EU of that capacity and it would be much more expensive, right?

They make an eu7000i fuel injected now, but you wont get 7.2k out of it. the i stands for inverter.