I've had exactly the same problem. My K12GT was overheating terribly and a visit to the dealer didn't solve anything. I did my own searches and traced around the net, this forum and managed to lay my hands on a servicing manual.
I was told by the dealer that a new radiator would be around £450 - (I think about $700?) - and another £400 to fit. I walked out in disgust. I stripped the radiator out to find bashed in fins and crud from about 10 years of no maintenance. I waited until the wife was out the door and shoved it in the dishwasher on full heat. It came out pretty clean but there is nothing I could do about those fins. (Just why BMW chose to put the rad on the floor right behind the front wheel and below the mudguard is anyone's guess. If K12GT ownership has taught me anything, its that "ze Germans" do not have all the best engineering!!)
I flushed the system and with some cooling system cleaner then refilled using the vacuum method. It solved the problem instantly. But, when I stopped, the bike puked small amounts of coolant. I put it down to overfilling.
All this work and I'd left a hose loose, which dumped all my coolant one day. (the British don't have all the best engineers either it seems!). So I refilled it, but this time I took particular care to make a really thorough job of the vacuum filling and bleeding. It never puked or overheated again.
1. BMW have put the radiator too low on the bike. This means that the rad gets smashed by small stones and road crud. It also means it's below the engine head so can't be filled by a normal gravity topping up method. Clean it out with a low pressure hose every time you clean the bike.
2. The fins are very thin and very prone to damage. The fins increase the cooling surface area of the rad so it's efficiency decreases when this happens. Also, the rad is prone to blocking because the vanes are so thin. Cruddy coolant must be avoided at all costs. A flush seemed to solve my problem - for now. A new rad will solve this, but BMW are under the impression that I have a bottomless wallet!!
3. I'm not sure how Honolulu got away with not vacuum filling. Perhaps the rest of the system is so efficient, small pockets of air can still be managed. When I put some real attention into this to make sure absolutely no air was consumed when I filled the system, it worked and never puked coolant again.
4. The rad cap isn't sprung on the K1200GT I've got. This is different to most other bikes and cars I've worked on. The idea being that the heat and pressure builds up, moves the spring up and allows coolant to overflow into the header tank. This is then recovered by the system and cycled back into the engine via the bottom of the tank. The rad cap sits above the overflow on the K12GT, meaning that there is a free passage to the overflow tank from the radiator. The amount of coolant in this tank (on the left of the bike) will show you if you need to top up the system but you cannot top up by just pouring water in the tank. The manual does not cover topping up coolant (a simple rider job in most other bikes) - you need to do it by vacuum filling, and that means dropping the old coolant and refilling using the vacuum method.
5. I've had a similar conversation as Honolulu with a BMW tech about refilling the system using the sidestand and just pouring coolant in. In my opinion this probably means if your cooling system is working perfectly (no bashed in fins on the radiator) you are getting away with a bit of air in the system. If you have a car or other bike have a look at your header tank or radiator filler and you'll see it's at one of the highest points of the engine. If you fill something up from the top, its likely the air will expel as you fill it, leaving the odd pocket of air to be expelled later. The K12GT fillers are not at the highest point. There will always be air at the top of the engine if its done this way. Bleeding from the points on the left will help, but it is unlikely to get it all out and this will (possibly) lead to overheating.
6. I don't think its your head gasket. If you are losing coolant (white smoke out the back of the exhaust) or white crud in your oil or vice versa then fair enough, but this doesn't seem to be your problem. Changing spark plugs is such a piece of major heart surgery (again a simple job on almost every other machine I've owned) its unlikely to have been done in a long time. I'd start there. Its a very normal and minor reason for an engine not firing on one cylinder and one I'd eliminate before I went for a head gasket.
7. It sounds like your bike is not cooling with its greatest efficiency, but it is cooling. The fan coming on when at a standstill is what the fan is for. If the fan is keeping the bike from overheating when at a standstill then it's working well. These bikes' cooling fins are enshrouded in plastic, meaning the air cooling is poor and water cooling is all they have when they're not moving. Another piece of amazing BMW engineering!
1. Your bike is probably cooling well enough. Like Honolulu says, let it settle down a bit and perhaps try refilling using the vacuum method again, but take particular care and bleed it well.
2. The cylinder not firing is more likely to be a simple spark plug long before the head gasket should be suspected. If you take the rad off and clean it, you can change them while you are in there. You could also do the valve clearances (another normally pretty easy job on any other machine), and refill the coolant.
3. My final point - controversial I know on a forum like this. These bikes were designed and built by nuggets!! There, I've said it! I've lost a lot of respect for BMW after owning this bike (my first beemer). The cooling system should be a simple maintenance task - just like filling the oil - but it isn't. The air cooling is nullified by the enshrouding plastic - there should at least be some sort of vent to allow air flow. The radiator is sited too low and vulnerable to damage. The cam chain is prone to jumping as a result of a lack lustre design in the tensioner - I've had to fit a manual tensioner and a guard as a precaution. The clutch squeels on take off - I am told because the oil pump drive wheel behind the clutch plates is too wide - so I need to retrofit a thinner drive from a later bike to see if that solves it when I change the clutch this winter. The tyre pressure sensors have batteries in them!! Batteries! For Pete's sake, I now have to remove the tyres to get at them so I can remove a permanent tyre pressure warning light. For some reason the headlight is high - this is because of a flimsy plastic adjuster at the back of the headlight which has broken off (another thread elsewhere is a great and ingenious solution to this problem). Spark plugs - difficult. Valve clearances - difficult. Screen adjustment runs via the engine management system - why!!? The fuel gauge is unreliable and used a strip which BMW know has caused problems - you can't trust it and have to go off the mileage! Why? You'd think with that lot that I've bought a real dud, but these are all problems common on this forum and impossible to pick up on a check before you buy.
But! It rides like an absolute dream. Over engineered yes. Difficult to maintain yes. But my goodness, if you can keep these bad boys going they are an amazing bike to own. I've just nailed 1700 miles in a week of touring in Scotland and would not have wanted to do it on anything else than my 07 K12GT. 60mpg - nearly 150 horses to pull me the wife and our luggage through the mountain passes and through the winding roads - excellent braking - an almost flat torque curve to deal with those changing bends and mucked up gear changes - comfortable seats - incredible handling (even two up) - ample space in the luggage - heated grips and seats ... it just goes on. If you can ignore the warning lights caused by batteries in the tyre pressure sensors, a less than perfect cooling system, judging fuel stops by mileage alone (as well as fuel efficiency), some other meaningless warnings (like a front light bulb, which is clearly still working) then I will recommend this bike to anyone. But if you can't use spanners and believe what BMW dealers say - and indeed believe that all the good engineers are German - get something else.