You will not get fogging once the developer hits the paper, but I suspect that there would be enough ambient light from the HID lamp to give you fogging while transfering the exposed paper from the contact printing frame to the developing tray.

BTW, the only reason I recommend leaving the lamp on for the entire printing session is because it takes about five minutes to reach peak output and just leaving it on avoids the problem as you just heat it up once. However, in your case, with the exposing lamp in the darkroom itself, it would probably be best to turn it off.