If it's the radiator fan, it won't have any effect in the engine as the pump won't be running.
R1 can get way higher, like 1M, so a much smaller capacitor will be required, around 220-330uF assuming the mosfet cutting out at 3V.
Just buy a 50A automotive relay, they're cheap and meant for this.
Think something else so it doesn't always trigger, imagine the fan starting everytime you simply put the contact for a second, not only it'll be annoying, it'll deplete the battery quickly.
I wouldn't trigger the fan unless the engine has been running for at least 20 minutes!
A MCU won't necessarily deplete the battery. It only needs to turn on with the switch contact, coun time (Or even better, read some emperature sensor and decide wether to run the fan or not).
In any case, it only needs to run for those 5 minutes. After that, it can release a transistor so the whole thing becomes unpowered.