There is no problem leaving the inverter on the batteries while you are charging the batteries. Everybody does that, as most inverters are "installed" so you can't disconnect them anyway.
Obviously, it is better to not have any draw on the batteries while you are trying to charge them as that slows down the time it takes (because your net amps in is less) and you want short as possible generator time.
Note you went from a 2000w inverter in the OP to a 1000w later. ???
The so-called standby draw of an inverter is fairly trivial in the scenario you describe. If you don't have a remote to inside the rig for the inverter, then if it means going out in the dark and rain to shut it off at night or to turn it on in the morning, then just leave it on
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1. 1991 Oakland 28DB Class C
on Ford E350-460-7.5 Gas EFI
Photo in Profile
2. 1991 Bighorn 9.5ft Truck Camper on 2003 Chev 2500HD 6.0 Gas
See Profile for Electronic set-ups for 1. and 2.