Rear fog washes out camera at night.Electrical question about reverse & fog light mod

[Funkdoobiest]

I installed the OEM rear fog light with dual filament functionality. But now it totally screws up the factory rear camera contrast at night. Like the bright red is overexposed and makes everything else dim and hard to see.

So this is my question as I sit here at work before I get into it later. I'd like for it to turn off with reverse light, I don't want a switch on it, I want it automatic. So do both wires going to the reverse lights get ground when off? If so, thinking I can splice the ground fog wire to the positive reverse wire. And when I hit reverse it'll turn to positive giving the fog double positive therefore shutting it off.

That's really the only thing I need to know, are both wires going to the reverse light grounded when not being used?