Update, or "How to come unscrewed":
After the 3/16 hex wrench finally was cranked on so much that it stripped the hole in the setscrew, and ditto for the 5mm I was advised to try next, I left it in frustration. BCS said I could drop in this week as we headed east and they'd drill it out - they apparently would try a reverse-flute bit. My brother said that's fine, but that I could probably do it myself. So yesterday I gave it a shot, sans a reverse bit.
The 3/16" cobalt bit followed the hex hole pretty much but eventually bogged down at the bottom trying to get through the hard stainless screw base. So I bit the bullet and bumped up to 1/4" bit, and it widened the hole but it also bogged down in the bottom where the flats were. I kept getting grindings of metal coming out, but it went on forever - the tip of the setscrew is pretty doggone thick. I put cutting fluid on the bits and slowed the bit speed and put pressure on it (a slow, oil-cooled bit runs cooler which means better cutting); tailings kept coming out, but no bottom. Aarrgghhh. I just didn't want to suddenly bust through the bottom and damage the mirror shaft somehow, so I kept stopping to check and vacuum shaving debris, and that's what took so long.
I kept looking for the tell-tale chrome of the mirror arm shaft at the hole bottom, but no cupie doll. Drill some, clean hole, flashlight - no bottom; oil bit, grind away; repeat for a good hour. I could tell I wasn't quite up against the thread edges of the setscrew/tapped hole interface, and I couldn't quite get a pick to start unwinding what was left in the thread path. So I elected to go one more bit larger and hoped I wouldn't grind away too much hole thread.
The 5/16 just fit inside, and with my tremor it was a bugger to line it up and press down without sashaying into some exposed outer threads in the hole, but I finally exposed some chrome at the bottom. I tried to pick loose some of the now doughnut-like screw bottom, but it was still intact and stubborn, and I was concerned because the pointed tip of the bit would dig into the chromed shaft. That wasn't working, so I used a 3/8 X 16 tap and it helped clear out remaining screw threads; but when it bottomed out on the mirror shaft, my forcing it a smidge more only ground out hole thread material along with the screw threads... oops. So I managed to turn the mirror 20 degrees so the bit tip wouldn't notch out a hole where a future set screw needed to get a bite, and pressed on with the 5/16 bit. At this point it was a relief to find the mirror actually turn far easier than the mirror on the passenger side after removing its non-problematic setscrews.
It only took a few turns and out popped the bulk of the remaining setscrew. I did a light cleaning with the tap, vacuumed out what I could, and used a pick on some remaining setscrew tip that had lodged between the shaft and the hole edges. I should have expected there to be a slight gap between the shaft and the sleeve it sits in, for turning allowance. There was some unavoidable dinging of the chrome surface of the shaft, so hopefully it won't corrode over time and freeze up the shaft. But boy had I reamed out the hole's threads! The new setscrew was sloppier than heck at the mouth of the hole, and I'm thinking, great... helicoil time.
But further down it bit fairly well, so I put a small dab of antisieze on the setscrew, moved the mirror back to the little white, innocuous, "permanent" alignment marks I'd put on before dispositioning it, and installed the screw. Needless to say, I did not crank down on either setscrew! Just snug. The mirror doesn't move without loosening them, yet it will be easy to get out of the way when negotiating the darned fence. Brother had recommended the red Permatex thread-lock stuff on the new screw, but I think he meant the blue version. Somehow using anything called "thread locker" just didn't sound real enticing; so I used anti-sieze (don't tell brother!).
But with the enlarged, wiped-out threads at the mouth of the hole, the original screw-in button cap would no longer grab. Even my new chromed metal pop-in ones wouldn't. But at least I could bend outward their springy metal "fingers" until they got a grip on what was left of the hole's thread collar. Hopefully the cap will stay put on the road, but I bought spares, and if need be I can just tweak the next one to a tighter fit.
So for those of you willing to somehow make it through all this rambling detail, which is likely only bored, retired engineers willing to settle for even the most mediocre entertainment, that's the story I'm stickin' to.
Joel