EE hat on:
(1) A "radiation shield" will help heaps - basically if you can keep direct sun off it to max extent sensibly possible. Reflected sun from bonnet (hood) and some re-radiation form other surfaces will happen but (2) handles that.
(2) "Forced" air cooling makes a massive difference.
A small fan with a very modest airflow directed appropriately will make a vast difference. You could probably even manage to arrange heat transfer from metal parts to modest fins for air to flow over. Fan can be vehicle powered - say a modest 12V "muffin" fan and a series resistor to slow it down. Probably no harm in running at full speed but sound level will drop into the inaudible range as speed drops. The sort of fan found in PC power supplies and case vents would probably be good. Those on processor heat-sinks are often slightly smaller and possibly busier. Brushless DC motor fans are not intended to be speed controlled by series resistor but the controller basically drives them as fast as it can with the available energy supply and by effectively reducing voltage with a series resistor you will get lower speed. This is not liable to hurt them for sensible speed reductions.
Try it at full speed first to see if you can tolerate the sound level.
Ensure that the fan will run when the car is left unattended, that it's days may be long on the face of the land. Sealed car with radiation and rising interior temperature could take a camera to 80C+. Or ALWAYS demount it (a nuisance as time lapse of pedestrians and of people stealing the camera could be fun.
A modest car battery is 20 Ah+ and a nice chunky one is 40 Ah +. If the fan draws 100 mA it will take 10% of a 20Ah battery in 20 hours. Acceptable. Run from its own 12V battery charged via adiode from the cigarette lighter if desired.
THe GoPro products are marvellous where their special features are useful or essential. If you need small, rugged, waterproof, not interfering with or adding to your more extreme attempts to harm yourself, ... the a GoPro may be the camera for you.
But, FWIW, in thius case I probably would not put a GoPro top of my shopping list.
They are excellent when the situation demands utter robustness but in terms of utility per $ and convenience of use and actual functionality any number of point and shoot cameras may be better and cheaper.
GoPro website here.
Look at the functionality and ease of use in your application. As long as a point and shoot survives and is not a physical or visual problem then can the GoPro match it photographically and in convenience? Partial list - view finder/LCD, zoom, controls, memory card access, download ... ?
If I was making that choice I may well end up with a GoPro - but maybe not.