Okay, I had this same hot start problem and it slowly got worse. So I bought the wire kit from BMW to fix the hot start situation. The dealers don't like to honor the free work for this because it takes 4 hours to do the job and bmw does not pay that many hours. It is hard to route the kit wires too. But with the same battery in the bike and the new wire kit it starts like a charm. I don't like to bad mouth dealer shops but I like to do my own work. I understand if someone does not have the tools, space or understanding to do mechanical work. Then you are at the dealer mercy. K12/13S bikes are the most expensive BMW bikes maintenance wise. If you can not step in to help keep your cash at home it can be nasty. And not all dealers are that good. When your gas tank is off you can see the alternator and starter. Pull the starter, open it up and clean everything.
Mount Vernon, WA.
tire guy, I'll mount your tires.