The 400w inverter might have come with two sets of wires. One for a cig plug and the other with clamps that go on the battery posts.
You should use the set with clamps on the battery bank and run a 120v extension cord to where the cooler is. Might need a rain cover if it will be outside.
A typical cig plug might only be good for 90 watts roughly. The thin wires will cause the inverter to "see" low voltage and shut down or start squealing its alarm.
The suggestion is to change out the 12v cig plug on the cooler for two fatter 12v wires to the battery bank and not use the inverter so it won't use as much battery to run the cooler. Depends on the situation/scenario whether that would be worth it.
Yes, "120" is "110". Household level "AC" power. The inverter makes 120 from 12. Use the "ten times rule" for amps draw from the battery. Cooler is 90 watts, then it will pull about 9 amps DC from the battery.
Two 6s is 225 AH and you can use 113 to get to 50% down. Means at 9 amps it will take 113/9 = about 12 hours to run the bank down to where it should be recharged, just from the cooler, never mind anything else 12v being used.
You need a plan for how to handle that, especially when the sun is not out much. Small generator?
