Andrew,
I had a similar problem with my 97 740il. Mine also stuck in 2nd and 3rd gear however a quick bump in manual mode seemed to do the trick after a few seconds delay. I was forewarned by my previous mechanic that this would likely happen at around 120K and that the only fix was a costly new trans. My car did not start having problems until about 140K. Upon my first time to my current mechanic (a sharp certified Porsche Master Mechanic) for a different repair on the car, he asked if I wanted him to fix my shifting problem also. I told him what my previous mechanic told me and he said I guess that’s why he’s your “previous” mechanic and we both laughed. He said he has fixed this on countless 7’s just by draining the trans (servicing the trans). Anyway, he changed the fluid in mine and it took about 2 weeks before it started working correctly I guess the new fluid had to dissolve the crud left by the old fluid. I’m at 175K now and have had no further problems. I drive the car very …spirited and use sport mode quite often. Good luck!