Update 1: removed right fairing and checked the coolant level in the radiator. It was 30 cc low, that's insignificant in the grand scheme. I think the coolant level is okay, unless there's an airlock in the system that I can't access without a full drain and vacuum refill.

I started and let the bike idle on the centerstand. It got up to 1/4 temp in about 4-5 minutes, then to 1/2 in about 8 minutes. That sounds like normal warmup behaviour and the fan came on as soon as it crept above halfway. The fan held the temp steady, with apparently an decent reserve (the rest of the gage).

Oil checked and added about 1/4 liter. No smoking gun there either.

The radiator was cleaned last August, for the first time in my 4.5 years ownership, it was not significantly obstructed and doesn't look bad now.

What's left?
The possibility that there was an intermittent failure of the fan (I hate intermittents).
Oil too low? A quarter of a quart shouldn't make the bike overheat, even two up.
Coolant too low? 30 cc refill is negligible.
Airlock in cooling system? I don't want to go through the drain and vacuum refill process, yet.

It seems that things are "normal". I will have try short, then longer rides to see if the trouble recurs.

 

Update 1: removed right fairing and checked the coolant level in the radiator. It was 30 cc low, that's insignificant in the grand scheme. I think the coolant level is okay, unless there's an airlock in the system that I can't access without a full drain and vacuum refill.

I started and let the bike idle on the centerstand. It got up to 1/4 temp in about 4-5 minutes, then to 1/2 in about 8 minutes. That sounds like normal warmup behaviour and the fan came on as soon as it crept above halfway. The fan held the temp steady, with apparently an decent reserve (the rest of the gage).

Oil checked and added about 1/4 liter. No smoking gun there either.

The radiator was cleaned last August, for the first time in my 4.5 years ownership, it was not significantly obstructed and doesn't look bad now.

What's left?
The possibility that there was an intermittent failure of the fan (I hate intermittents).
Oil too low? A quarter of a quart shouldn't make the bike overheat, even two up.
Coolant too low? 30 cc refill is negligible.
Airlock in cooling system? I don't want to go through the drain and vacuum refill process, yet.

It seems that things are "normal". I will have try short, then longer rides to see if the trouble recurs.

I had a similar problem with my '05 K21S and it turned out to be air in the water pump. There is a bleed nipple on the left side just above the water pump. First, connect a piece of clear tubing from the nipple to a container and then with the engine at normal operating temp., gradually open the nipple and keep it open until there is a steady flow of coolant. Hopefully this resolves your problem.

 

Here's an idea, shot in the dark really but it fits with your scenario. If the mixture of your coolant is incorrect, too much water and not enough glycol, it may have started to boil creating an air pocket leading to your overheating condition. Have a shop check the specific gravity with a tester to see if the mix is within spec.

 

The thermostats in the k bikes are problematic. Sounds likely if the bike overheated so quickly.

 

Drain and refill the coolant carefully under vacuum doing the thermostat and water pump bleeds on the side stand. Then go from there.

 

CLOSURE, I HOPE..

Short version: I believe the overheat was caused by low coolant level. It wasn't the dashboard warning triangle inadvertently indicating the emergency flasher. Thanks to all who suggested various approaches.

There are four ports into the cooling system: 1 (right side) radiator fill cap, 2 (left side) 15mm plastic vent screw on the radiator top hose, 3 (left side) 11 mm bleed nipple on the coolant pump, 4 overflow tank vent to ground. All of them are at the same distance from the ground with the bike on centerstand, meaning there's no single high point from which to bleed air.

I removed the right side fairing and filled the radiator, it took 30cc, negligible. I warmed the bike, added 1/3 liter oil, again negligible. The bike ran and fan came on and off. At this point it works but overheating was unlikely to be resolved.

I GET that BMW calls for vacuum bleeding. Drawing almost all the air from the cooling system, then replacing the vacuum with coolant will fill all the little nooks and crannies. I have the tool to do it and have done it that way, once. Bit of a hassle. Whether it requires to be done is another matter, take it up with Ricardo Engineering and BMW Motorrad.

I filled the overflow tank which was about dry. I plugged the overflow line down at the kickstand with a golf tee and used some bits from my Mityvac brake bleeder to rig a way to feed 50/50 BMW blue coolant into the expansion tank, with the plastic bleeder screw and coolant pump bleeders open. Closed those taps when coolant runs out and the system is now full except for traps in the cylinder head, which some claim don't actually exist, or can be ignored. I take no position on their existence and they haven't questioned mine.

Closed bleeder screw, pump nipple, overflow tank and ran the bike with a catch can under the overflow line. It collected about a cup of coolant; the temp gage went up to 1/2, the fan cycled on and off over some 15 minutes. Put a clear line on the coolant bleeder, opened it and only hot coolant came out.

Having refilled the entire cooling system and gotten it to bleed as intended, a test ride of 7 miles in the immediate neighborhood (about 400 vertical feet) seems to confirm I'm okay. Ambient air temp and humidity are mid to upper 70s, temp gage didn't get to halfway, fan stayed off.

Moral of the story: a little knowledge is a dangerous thing. Sure, something major can go bad, but MOST LIKELY it will be some little thing, if you catch it early on. In this case my failure to check coolant level for the last six months was the problem. I think very gradual coolant loss is caused by the tiny bit of coolant splashed into the overflow hose at each bump in the road, which then dribbles out the vent hose a fraction of a cc at a time.