I had the same question in my mind, "will the alternator handle a load on the inverter and still charge batteries"
I just installed a 1200 Watt inverter mainly to run the GAS/ELECTRIC refrigerator while on the road (instead of using propane) Hey, the televisions "might" accidentally turn themselves on also
With the alternator (and engine) running I get 14.4 volts on both the chassis and coach 12v circuits. When I turn on the inverter with no load, still get 14.4 on the coach circuit. Add a television, no change. When I turn the refrigerator on with the propane valve off (forcing it to run on 120v AC) the coach circuit drops to 13.6-13.7, while the chassis circuit stays at 14.4v.
At 13.6-13.7 on the coach circuit, my thinking is that the coach batteries will still charge, while the alternator is providing the power for the inverter and the fridge.
If my thinking is flawed (or downright off the tracks) let me know....