I'm with Drew. Our inverter has a built in charger and transfer system. When the shore power is turned off, inverter automatically takes over in milliseconds. When shore power is re-connected, inverter turns off in 10 seconds or so and AC restored to shore power. When AC is connected, if the battery is low, it will be charged automatically.
Charge controllers work happily in this environment. The only thing I worry about is total charge current when AC is connected and also charging. I haven't had any issues yet, but since I can set the charging current on the inverter, I just limit it to about 60 Amps.
I'm thinking of adding a Sterling-Power battery to battery charge controller to be able to charge the house batteries from the bus alternator.. ( Sterling Power 60A Battery to Battery Charger
). Note that you need to have this kind of system if you're charging the batteries from three different sources like this.
Here's my electrical layout (at least the DC side):