120V side:
It looks like at 120 Volts A/C, the 10 Gauge wire from the transfer switch to the inverter than from the inverter to the circuit panel should do the trick.
12V side:
This is where things get trickier. Since I am planning on installing lithium batteries right beside the inverter, I am estimating that I will need less than 4 feet of cable from the inverter to the batteries and back, which looks like 4 guage will work. I could use 2 guage if I wanted more insurance policy. It appears that the length of cable can play major factor.
I am referencing this link to determine which gauge of wire to use.
12 Volt Wiring: Wire Gauge to Amps | Offroaders.com
I am open to suggestions and feedback. Is there a favorite Lithium batter out there?