Looks like I fixed the problem.
I rewired the control box (+) to the positive wire of the cigarette lighter and the (-) to the negative wire of the lighter. I also separated the bundle of laser head signal wires from the 8-guage power wires running my amplifiers. After taking it for a test drive, I no longer got any of the false alerts that usually appeared with each turn I made. I got no alerts of any kind, but that's ok since I did not catch sight of any laser sources. I'll know if it's working for sure on my ride home from work today.
In doing this, I relieved myself of the stress from thinking my new equipment was malfunctioning or possibly damaged, but also discovered something about my car that really pi$$ed me off. I originally grounded the control box to a steel block inside the fuse access panel. This block is what the actual fuse box is mounted to and it also has an easily accessible hole for grounding. It is even labeled "COM". My car alarm and stereo are also grounded here.
The problem? This block is totally brown from oxidation/rust. I tested the resistance between the block and a bolt on the actual frame of the car, and the resistance was .7 ohms. Almost an entire ohm. Given that the multimeter itself reads .1 ohm when the leads are connected together, and that the resistance between any two bolts on the frame also read .1 ohm, I concluded that this block lost some of its conductivity due to the rust. You would think that if they wanted that to be used as a ground, they would have constructed it out of higher grade steel that would not so easily rust and was better mated to the frame of the car.
Now I'm starting to wonder if this apparent grounding loop is what started all this mess for me in the first place. Before purchasing the LPP, I had an Escort ZR3 that started falsing for what seemed no reason at all. The heads for the ZR3 were originally damaged by a mechanic, and replaced at his expense. Later, false alerts from the ZR3 heads and the 8500 itself started appearing on the 8500 display when hitting high RPMs, even when the ZR3 function was turned off. Thinking that my ZR3 system was toasted, I got frustrated and removed the heads. The 8500, still connected to the junction box, continued laser falsing at high RPMs. I turned its laser function off, and, a few months later, got my first speeding ticket in more than 3 years. Fortunately, that drove me to place the LPP in my car 2 weeks later.
I'll try grounding the ZR3 junction box to the same spot as my LPP. If that works, I'll then have high and low protection against laser again.