I have not had good luck charging directly with any "smallish/cheaper" generators even if they are 120/240. The EG4 inverters seem to be fairly picky and require a very clean sine wave and stable voltage. This means you might as well use a cheap generator and a chargeverter, or any similar 120vAC->48vDC charger direct to battery to get enough charge to get thru the night.
I typically tell off-grid clients to get in the habit of checking battery voltage at dusk before cooking dinner.... and if SOC% is <50%, then fire up the Genny, and run it until bedtime.