I’d like to make the same swap, but I’d read that BMW said you needed to also swap the K12S clutch for a K13S clutch to make it work. Not true?
Since I now know the answer, I am re-posting to help anyone else with the same question.
I bought a used trans from a K13GT. It has the larger holes drilled in the shaft to bathe the clutch in oil (the 12S clutches were notoriously under-oiled), so when looking for salvage, look for a K13 trans, NOT an old K12 trans, because you'll just be buying the same problem set (an underoiled "wet" clutch which imitates a dry one, and transmission dogs which are NOT undercut, resulting in common failures of 2d, and sometimes 3rd, gear). The K13 iteration fixed both those issues.
However, when you buy a salvage (or new, but why would you buy new for a bike this old?) trans, it will come without the little spinning collar which sits over the shaft and also has two holes drilled into it and throws out oil to wet the clutch. Unless you also buy that part from a K13 (and if you do, it WILL fit a K12), you will need to drill out (and emery to baby's butt smooth) the two, smaller holes on your K12 clutch shaft collar to the same diameter as the holes in the K13 transmission shaft. I found this out before doing the work from a BMW mechanic of 40 years, and the advice was spot-on.