Here's a side view of the fuse panel removed from the lower housing, and with the cover removed.
The user-accessible relays and fuses are on the bottom side in the pic, There's a fence about ⅜" high around the bottom side which hides the fuses. Not counting the fence, this thing is about 1.84" thick, plenty thick to house a layer of relays and other stuff.
Examining it, it appears to be snapped together with plastic clips/barbs. As before I'm hesitant to try and separate it without the right tools and a little more knowledge
I looked around on the web for more info, but struck out in my brief attempt.
Anyway, to restate the obvious, I suspect there's a bad relay housed inside this thing which is keeping the fog lights on.
If this section can be separated without damaging it, I'm with the others - de-soldering a relay and replacing it is not very difficult given the proper tools. I get by with an old pump-and-click solder sucker, so it takes a few attempts to get all the solder off one of the relay pins, but I can usually get them out without too much trouble, and without killing the ring around the plated-thru hole on the board. A powered solder sucker is preferred but that's usually something seen in bigger electronics shops than my garage
In summary, I think this beast needs to be dis-assembled to fix the fog light problem. An alternative hack is to wire up a new relay externally to the box, but I'd try to dis-assemble the fuse panel first before opening that can of worms.