Setting up a solar system is very straight forward
Solar panel----Solar Controller----Battery----Load (See note)
The load may also be connected to the controller depending on the max current draw.. The configuration above is in my "Living room" running a radio.
It is the one the battery maker suggested
I would recommend a LiFePo4 battery for reasons including the one you stated.
Advantage of connecting load to controller: Low voltage protection
Advantage of LiFePo4.. You might not need low voltage protection
Disadvantage of solar: If battery is not big enough cooler may cease to cool due to low voltage. This happens with LiFePo-4 very late in the discharge cycle (like below 10%)
