Elaborating in really rough numbers, a 100 amp hour 12 V battery has 1200 watts available and in the case of lead acid, only 50% should be used if the battery is to last more than one or two seasons.. In a nutshell, (again rough numbers) the cooler alone will use up a 100 amp hour lead acid deep cycle battery in one day. That is if not much else is running like charging phones, computers, autopilot, chart plotter.. lights, fans, etc. .. If your battery bank is LiFe, the general guidance is 80-90% depletion is OK so for each 100 amp hours of capacity, you have around 1 KW available so the cooler would run around two days . That again is really rough SWAG stuff and accounts for nothing else running. The cooler duty cycle , depending on ambient temperature in the boat, may be 20-30% more or less than these numbers..
A minimal sized solar panel of 100+ watts would extend the battery life further, but anything less than 100 watts is not going to help much.
Good luck !