I do not think there is one right answer to this question.
On my coach...my inverter is wired directly to the batteries since I have a residential refrigerator...
Flipping the battery disconnect switch does NOT turn off my fridge since my fridge gets powered from the inverter which is making AC from the house batteries...so for my setup...having the battery disconnects disconnected and my AC cord plugged in would still charge my batteries...
It all depends on how your inverter/charger is wired.
When you flip the disconnect switches...does that kill your inverter?...I would assume it would IF your inverter is on the opposite side of the switch from the batteries...
Like I said...on mine...when I flip the switches OFF to disconnect the batteries...my fridge will STILL work and my inverter/charger STILL has power going to it...so I know which side of the switch my inverter/charger is.
Good luck,
joe.
2007 Tiffin Phaeton 40 QSH 350Cat (3 women and 4 slides...Just RIGHT! :B)
2012 Jeep Liberty Sport (toad)
2005 Ford Excursion V10 w/4.30's (Sold)