Is it dangerous to run electrical cords through heating vents?
Any electrical socket in a bathroom must have GFCI protection. You're damp, you touch something with a ground fault, feel a slight tickle and wake up wearing a halo and wings. Of all places to hack together power supplies, a bathroom is absolutely the worst place to do this. Mystical theories about the heater protecting the strip from dampness don't remove the shock hazard.

Fire hazard is not your problem here