I don’t know if it is fielded yet, but in fact subs can now launch AA missiles, though generally not to the altitude of TWA800:
https://en.wikipedia.org/wiki/IDAS_(missile)
The US system is more advanced, though I vaguely recall it is ejected out of somewhere extremely unlikely, like the garbage ejector.
Follow up:
Those sub AAM systems were not operational at the time of TWA800.
The Russians launch theirs out of the torpedo tube. The rocket is encased in a waterproof shell the same size/shape as a torpedo. It pops up to the surface and the shell is blasted away as the rocket engine ignites.