Assistance with circuit design
Wire your loads in parallel instead of in series.

Make sure your temperature switch can handle the current required.

ADD A FUSE ON YOUR BATTERY DAMN!!

Batteries are excellent for starting fires. A 12V lead acid battery will happily put out enough current to heat your wires red hot, melt them, and set everything on fire.

Add an overtemperature safety. When one fan fails, or someone covers the "do not cover" heat sink, or dust accumulates, etc... the Peltier will overheat and destroy itself. It is not cheap.

Add an undervoltage lockout. Discharging lead-acid batteries too deep will very quickly render them unusable. Check minimum allowable voltage for your battery.

Now, you have 3 inputs to your switch:


Fridge temperature: run
Peltier too hot: stop
Battery low: stop


It would be a bit complex to wire all this with relays, so I suggest keeping your mechanical temperature switch (if you already bought it...) and wire a dead simple safety for the rest: a low-RdsON MOSFET, a LM339   comparator, TL431 for low-battery check, and your choice of thermistor.

Test the setup using a current-limited bench power supply, not your beefy battery.

EDIT: I upvoted your question to compensate for the -1 you had because, although you clearly gonna need lots of help, you took the time to explain your problem clearly...