I've never done the math, but i bet the alternator won't keep the batteries charged as fast as the heater and inverter draw them down.
Max electric heater by law is 1500 watts. Assuming 100% conversion efficiency, that's 125 amps of 12 volt power. Not sure what the rest of the coach and chassis draw, but that's a lot of amps to push at 12 V to me. It's diesel fuel either way, generator or engine. We use the generator.
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2015 SOB 5th Wheel
2013 Chevrolet 3500 CC, DD SRW
1 Wife, 2 Kids under 5
Camping nights: 2015 - 19, 2014 - 39, 2013 - 48