The double sided 3M tape that comes with dash cams is extremely solid and I don’t see how this would be a safety issue. I’ve had to remove it several times and it always requires a heat gun to get off.
The DE Tech lead will ultimately determine whether this suffices as a safe way of securing a dash cam. If it’s deemed unsafe, you can always just detach the dash cam for the track event.
In terms of wiring the dash cam in, look up the Meknic hardware kit on Amazon. It costs less than $20, works with all dash cams and makes this job a breeze. I’ve used several of these over the years, and they work very well.