Does your inverter charger have an automatic transfer switch? If not, you probably want to put one in. This senses 110V AC, and bypasses the inverter when on 110V AC power.
I have my inverter charger set up a little different. The 110 side feed thru the panel with the main 30 amp breaker, and goes only to the "outlets", and has a built in transfer switch. I also have the microwave, air conditioning, water heater, and battery charger (house bank) excluded from the inverter. (My boat has specific breakers for each of these loads).
The inverter has its own battery bank. A 2000 watt inverter should have at least 100 amp hours of usable battery power dedicated it. (this means two group 31 batteries). We use 2 100 amp hour LI FePO4batteries, with a usable capacity of about 180 amp hours.
The 2000 watt inverter will suck down the "normal" group 24 battery which comes with the boat in very short order.
My particular inverter will also sync with the mains power, and if the voltage is low, it will boost it to 120 volts, up to 2000 watts.
The size of cable depends on how far your battery is from the inverter. I use 2/0 and it is about 40" round trip (20" each way, fused at 300 amps 12 volts.
You need to have sophisticated type of battery monitoring, to determine, voltage, amps in, amps out and state of charge. for the inverter batteries. The Victron 702 is probably the Best Buy and choice.