...after I advanced a level in trading, the small window with my character stats on Character Info station tab still says 9665/9500 for trading XP, even after /updatestation.

If I try to take the license mission again, it shows correct value - 9665/16250.

Even after you exit/reenter the station?

/updatestation will not update your charstats.
They only get updated when you enter the station/login.