If such a mountain range existed and if it had a suitable spot for an observatory (size, topography, not plagued by bad weather, etc.) a possible way of building and operating it would be by extensive use of RC robots. You could stay at some cozy room, coffee mug at hand's reach while the robots are working and freezing their butts, gears, wheels, whatever, over there.
But why would you do that vs sending a telescope into space? I mean, for "good" weather we are talking about permanent glaciers, -80F and worse, 200+mph winds, etc. We already have plenty of mountains taller than our current observatory altitudes. We don't go much higher for a reason. After a certain point, the cost-benefit swings towards space. If you've ever been in the Himalayas, you'd know how often those mountains are socked in with huge weather systems, which doesn't equal good viewing conditions, I would suspect. Trying to engineer systems that can be operated remotely, yet withstand 200mph winds, temps at -80F, etc, while theoretically possible, is, well, optimistic?
There is a weather station at the top of Mt. Washington, a lowly peak at 6288ft, but with a micro-climate that approximates the higher peaks. I've been on it at -70F wind chill, for example. I've been blown off my feet, pelted by very large chunks of ice launched by the wind, etc. It's a dangerous and difficult environment in the winter. The observatory is staffed year round, and staff members have to regularly go outside to maintain equipment. Google will quickly provide images of rime ice, such as here:
http://www.ultimatechase.com/chase_accounts/mount_washington_summit.htm
So, we'd likely have a dome covered with tons of rime ice and solid ice, the dome and telescope being assaulted by huge chunks of ice at 150mph+, snow drifts covering the entire side of the buildings, frequent white outs, etc. We can hand wave that all away with "robots", but the engineering challenge rather exceeds our current capabilities. Why spend all that money for such limited viewing opportunities when we can just toss a scope into space?
I'm not a meteorologist, but I doubt that any 12km peak would have nicer conditions than that - certainly the existing 8km peaks have equivalent to worse conditions. Perhaps some super plateau like in Chile, only much higher, would have milder conditions, but the OP is about mountain ranges, not plateaus.