Okay so here's a small update. Long story short, still no luck.

However, I can confirm that I have switched voltage between the coil pins. 0V when low beams off and fog light switch off. Low beams on and switch I get anywhere from 9-11 volts (seems low?)

And as mentioned in my original post, I have constant 12v across two horizontal pins and when bridged, fog lights come on.

So, I turned my attention to the relay. When I apply 12v to top and bottom horizontal pins, I get an audible click and continuity between the switched (vertical) pins. Take 12v away and the continuity goes away, as expected. So, the relay seems functional. But, when it's plugged into the slot, it does not click when low beams are on and fog light switch is on.

So, it seems like although I have measurable voltage change between the coil in response to the fog light switch, the coil is not activating the switched side of the relay. I also cleaned the contacts of any dust/debris so I'm hoping that it's actually making contact.

Maybe I can try opening the top of the plastic relay housing and see if I can access leads on the coil side of the relay to see if it's making contact

A question for more electrically savvy folks out there, could a lower voltage on the coil side (when compared to the voltage on the switched circuit) impact whether or not the coil can activate the switch? Or maybe current on the coil side? Just spit ballin' here.

And thank you all for the help so far, it's truly appreciated.