I had a single channel SG camera that was funny like that too, but i resolved it by re-flashing the firmware ( at that time i don't think i had flashed it once )
you can do some simulated power up / down in the house with the front camera unit, and a wall USB charger if you have that, that is what i do testing cameras on my computer table.
Have you tried to "jiggle" the power plug slightly in the cigarette lighter socket when it have failed to start, you could have a issue with the power source too, that get momentary solved unplugging and plugging back in.
The factory SG power supplies normally do fine, but even the best can make a boo - boo now and then.
Maybe also for a while try to give the 12 V plug a push into the socket before you turn the key, if that solve the problem it might be a seating problem there, or a build problem in the same place.
I have experienced plugs coming loose myself, but this was in 3 X extenders and not my car factory plug on the center console, but the 3 SG power supplies i currently have plugged into that 3 X extender work flawless.
Don't think you do anything wrong, more inclined to think you have gotten a gremlin in the deal or one pop up shortly after, either way SG dont tolerate those critters in their gear, so we just need to figure out what make most sense to do.
Replace your PSU - replace the whole package ASO
As Jon say memory card could also be the culprit, i have owned memory cards that have died on me scary fast, so that can happen.
So if you could do some static testing at home with just the front unit on a wall charger, that would be nice. ( simulate start ( stop by plugin in the USB plug and unplug it a little later from the camera, i would try 5 minutes on - 5 minutes off )
If you don't have that option, try in the car without the rear camera connected for a few days and see how that go, if that work fine it might be the cable to the rear camera or the rear camera itself.
As Jon say you can try to slide the rear camera off its mount, use the other cable in the package and then just put that and the rear camera plugged in over the sun visor to have it out of the way when driving.
Also as Jon say, if you have another memory card to try, almost anything will do, try that and see how that go. If you don't have a spare card maybe run H2testw on the card that came with the camera using a PC and the provided little card reader.
A little annoying, but the guys will get you sorted.