Backup Lamps out after backup cam install

2006 Grand Cherokee Laredo 4x4 3.7V6
So I installed a backup cam. I spent 3 days meticulously putting this thing in. I tested six times throughout the process and the camera worked fine: put it in Reverse and the image appears on the screen. I even mounted and adjusted the camera. Everything worked fine. I was finished and put all the panels back in the car. While i was testing the lights to make sure nothing was out my daughter noticed the backup lamps came on for a second and immediately went out.

Now, I no longer have backup lights and I don't get a video feed trigger (so no power).

I researched my butt off trying to figure what happened. I searched for a fuse for the lamps but there isn't one besides the main lamp fuse, but all of the other rear lights work. So I thought it might be in the transmission.

I took it to the dealer and paid $100 to diagnose the problem. They said it is the Front Control Module, but only the part that deals with the back up lamps. They also said there is no fuse for the FCM. I now have to pay $650 to replace the entire FCM. I can't believe a fuse doesn't protect the FCM!

Does this sound right? A tiny backup camera can blow PART of a computer? This camera can't draw much power.