The factory wiring uses the body control module to switch the fog lamps on/off. The factory relay is on the inside of the junction box (fuse panel) under the cover on the drivers end of the dash. There are also 2 spare relays in that panel.
The way it's designed to work is that you have the vehicle running, Turn on the headlights and select the fog lamps on with the fog switch. This tells the BCM to turn on the lights and the fog lamps. If the headlights are on low beam the fogs will be turned on by the BCM through the fog lamp relay. If you select high beam the fogs turn off. Without the BCM in the chain the lamps will not stay on or switch off with the high beams.
Power comes from fuse 9 under the hood, into the junction box and fuse 19, comes out of fuse 19 to the relays power feeds. The relay gets turned on/off on the ground side by the BCM. When on power comes out of the distribution center on pin 33 of connector C3 through a light blue wire. That feeds down to a splice to feed the two lamps.
(Fuse 19 also feeds the trailer brake light relay in the back if you have the tow package)
The multi-function switch only tells the BCM that you want the fog lamps on. And unless the BCM is the upscale one you need to program it to turn them on as well.
If I was adding fogs to a Libby that didn't have them. I wouldn't use any of the OEM interior pieces. I would add a relay under the hood, a switch on the dash and have the power to the relay switched with the low beam headlight.