I had an issue with my LED tail lights, when I pressed on the brakes, my front parking lights came on. Upon the conclusion of my investigation, I found that the regular bulbs have 2 filaments, one for tail and one for brake. This prevents any interaction between the brake lights and tail lights. My LED bulbs used the same circuit for both, and merely adjusted the brightness depending if they had voltage on one line vs two. That created a feedback problem into the taillights/parking lights whenever I stepped on the brakes. It first came to my attention that something was wrong when I was parked, car shut off, and the alarm to tell me I left my lights on started going off every time I pushed on my brakes. When I was in the garage, I could see the front parking lights/blinker/hazards come on as I depressed the brake pedal.
I haven't tried using cruise control since installing them, but I would guess that your problem is similar to mine, except that your cruise control is reading feedback from the tail lights into the brake light circuit. I. E., when you have your lights on, the cruise control thinks you are riding the brake, and shutting off.