Are the symptoms the same for the CC since '05 or have they changed ?
The reason I ask is the brake lamps being LED without load resistors might be a low enough value that the CC stops working ( need to have operational brake lamps to have the CC work ).
The only other external item is the CC disconnect switch, it is on the top of the brake pedal lever. Quick check of this is to turn the key to the run position and meter both sides of the disconnect switch. Light Blue w/ pink stripe wire is from fuse # 14, Black w/ yellow stripe goes to the PCM. With the brake off, the switch should be closed, and + VDC should be on both sides.
The reverse circuit, the output of the DTR ( Digital Transmission Range ) Sensor activates not only the reverse lamps on the truck, but also the PAM ( Park Aid module ) and the trailer tow reversing lamp relay coil.
At the PAM ( above the parking brake lever behind the dash ) the wire from the splice after the DTR is the Dark Green w/ Yellow Stripe wire.
Parking brake ( and other safety items, wheel chock, etc ) , turn key from off to run ( no need to start ) and put in reverse. The Dark Green w/ Yellow stripe wire should have + VDC on it.
Until you verify the cause of no reverse lamps, and test the reverse circuit, no need to go looking at anything else.