LiIon cells may not have enough voltage to run a dashcam- a few better ones will reach 4.35V fully charged, most top out around 4.2V, and there's a significant rate of voltage drop during use with all of them. A pair wired in series run through a "buck driver" (like powerbanks do) would work but running LiIon in series has it's own issues which can be downright dangerous unless dealt with properly. Running several cells in parallel would slow the voltage drop and might give a decent runtime. I'd feel a lot better if you used a large LiPo battery- they are much safer and with a more steady voltage in use. However this might work and I'm interested in your results if you try it.
As far as heat in a closed car and LiIon, I wouldn't worry about any danger. I recently suffered a car fire and had a LiIon flashlight inside which I couldn't retrieve. It took several (around six) minutes of estimated 800F-1200F degrees before I saw the light explode. Another flashlight enthusiast tested this scenario with a gas torch and came to a similar conclusion. Heat as high as cars see inside will degrade a LiIon cell though, shortening it's useful life appreciably and fully charged LiIon cells stored or used in such heat need careful monitoring during charging where most problems will show up. I wouldn't charge LiIon in-car because if anything goes seriously wrong you've got a potentially lethal Hydrogen Flouride gas poisoning probem at best and a "venting with flames" or cell explosion scenario taking the car with it at worst.
Never use cheap low quality LiIon cells or chargers and know the dangers- these are NOT like any other batteries at all.
Phil