Yes you can use the gen to supply the 120 for the converter--but the gen needs house batteries to start it AFAIK and there is a question about those batteries.
On the fourth day you lost pedestal power so no 120v to anything including the converter so no 12v output from converter. Started gen and now have 120v to run the air but still no 12v so converter did not come back on. Started engine and now you have 12v from alternator.
Whatever knocked out the pedestal also knocked out the converter--it has protection fuses, usually a pair of 30a. Replace both and see if that fixes it. it could also be the input breaker as suggested by others. It could also be that the converter is plugged into a receptacle on a string that has its GFCI popped.
However the batteries are very weak as seen by dim lights when the converter was off. (I ASSume the "full battery" indicator at home is because plugged in.)
The "monitor" will show full batteries even if they are dead as long as it "sees" 12.7 or more volts, which the converter supplies at 13.2v or so, 13.6 being usual.
Your real problem is why are the batteries so weak even though you plug in at home and the converter should be maintaining them.