The cooling system needs 200W to operate (and maybe a bit more for compressor start) in 120Volt mode. It may draw less if in 12V mode. If it does need 200W to operate, that is about 17 amps. you would need at least a AWG 12 wire for that ampacity.
Yes, you can connect another battery and you should. A typical marine hybrid battery has a capacity of 85 AH, 42AH of that is usable and then the battery needs to be recharged. So, using one battery, you could operate the cooler for 2.5 hours continuously. A second battery will double that to 5 hrs. If you have a compressor driven chiller, it will have a duty cycle (not run continuously). That may be as low as 1/4. You may get 20 hrs of run time with two fully charged 85 AH batteries.
For more direct advice, post what you have and particulars of what you want to do.