Having ruled out all the obvious, my first guess would have been the multi function switch itself. On mine, if I use the turn signals and then push it back to the neutral position manually, my left brake light wont work until I jiggle the handle. If I use the turn signals and have it come back on its own after making a turn, it works fine. So the switches can definitely cause some funky problems. Though I have to admit, your symptoms sound an awful lot like when my flasher died (the round part on the fuse box.) I would try a heavy duty one see if it makes a difference. Let us know what you find.
94 YJ - New frame, 3.5 inch Superlift leafs, 2 inch body lift, 33x12.50 muddies - 4.0 5 speed
95 Mustang GT - Too much to list - 5.0 5 speed
01 Yamaha Raptor 660 / 02 Warrior 350
A couple other Ford products