Your bike is unlikely to have a factory fitted BMW immobiliser.
My bike is the same year and I had a Datatool alarm/immobiliser system 3 fitted 5 or so years ago. They are quite common, although there is another popular brand out there. When you go into bike dealers, they are the ones that you hear peeping all day since they put the system into 'service mode' so they can move the bike around and remove the battery etc for charging. When the bike ignition is turned off, there is a red led in the rear light that should flash every second when this alarm is armed. The Datatool alarm should come with 2 wireless key fobs which are 'paired' to the bike. Ask the bike seller about it.
I have just labotomised my alarm to circuit board level with zilch help or info from Datatool so now I know their secrets, how it is wired on the bike, how it works and in my case how to fix it! You will not find this info on the internet and I won't publish it, since it could compromise the security for other owners. If you are serious and get the bike, send me a private message with your address and phone number and I'll talk turkey.
If the alarm is cutting out on the bike and it is a Datatool system 3, I would expect the alarm to become active and making a very loud noise with the indicators flashing!
There are other possibilities: ignition switch, stand cutout, engine cutout, fuel pump shut off, fuel filter blocked. A Datatool alarm does make connections in some of these key areas and a badly wired alarm/immobiliser may still be the culprit. But you would need to provide more information.
If you are not happy or confident fault finding electronics and wiring, you should factor in a rough estimate on the price of the bike. From what you say, if the rest of the bike is ok with no oil leaks etc it is down to fault finding labor costs. Because Datatool like to keep their secrets to themselves and their installers, you may have to get in touch with one of their local dealer/installers. If you just go and start chopping wires and remove their unit, you will have a non running bike - so don't even think about it! Many haven't taken the trouble or don't have the knowledge to fix these alarms and have just got fed up and put the bike back to a no alarm stock. That could take at least a couple of hours labor to do a proper neat job. The benefit of reduced insurance is now small on an older bike unless you are a young rider with lots of excess. However my Datatool has only had one fault in over 5 years and it is quite convenient so I quite like it. The problem with mine and probably many others can start when you replace a battery on the bike and their sensitive circuits locks up. I was ready to trash mine, but now it's OK.
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