Drawing 125 amps from a 200AH battery bank will get you that voltage drop for sure and knock out the inverter for low voltage. Needs more AH in the bank. Also the wiring from inverter to bank should be short and fat to reduce voltage drop just from the wiring.
You can use the inverter and shore power (or generator) the same time. Say you only have a 15a shore power connection. You can run the high amp things on inverter, but plug your charger into the 15a shore power so it is not on inverter making that loop, but does support the battery bank. Depends on the charger's amp size if it will cover all the draw, but it helps. it keeps charging between high amp events, so the battery will catch up over time.
Not clear about your inside and outside outlets/receptacles, but don't run the charger off the inverter to the same battery bank the inverter is using. That is the loop that drains the battery because the charger pulls more amps than it puts in.
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.