I like bass clarinet. I cannot think of any classical clarinet piece that like, but I enjoy baroque music by Telemann or Graupner written for the chalumeau - we have an excellent player of that instrument here in the area.
I indeed hate the altissimo register, that's why I enjoy jazz players who rarely use it, like Jimmy Giuffre, or with a very individual note, Pee Wee Russell, or Edmond Hall.
Re the thread title: A classical music instrument is an instrument used in classical music, period. Not exclusively, of course.