#Pyth, 53 bytes

u+GO?H+W<K/G\B3+W<Jl@"SF"G2\F\S\B+WqK3+WgJ2\H\S\B_UQ[

Try it online here.

This feels way too long, I think another approach may be required.

u+GO?H+W<K/G\B3+W<Jl@"SF"G2\F\S\B+WqK3+WgJ2\H\S\B_UQ[   Implicit: Q=eval(input())
                                                 _UQ    Reversed range from Q-1 to 0
u                                                   [   Reduce the above, with initial value G=[], next value as H:
                    @"SF"G                                Keep elements of G which are in "SF"
                   l                                      Length of the above
                  J                                       Store in J - this is the number of strikes and fouls so far
          /G\B                                            Count number of "B"s in G
         K                                                Store in K - this is the number of balls so far
    ?H                                                    If H is not 0 (i.e. not final pitch):
                           \F                               Start with "F" (foul is always available in non-final pitch)
                W<J       2                                 If J<2...
               +             \S                             ... append "S"
       W<K    3                                             If K<3...
      +                        \B                           ... append "B"
                                                          Else:
                                           \H               Start with "H" (hit is always available in final pitch)
                                       WgJ2                 If J >= 2...
                                      +      \S             ... append "S"
                                  WqK3                      If K == 3...
                                 +             \B           ... append "B"
   O                                                      Choose one element at random from the available options
 +G                                                       Append the above to G
                                                        Implicit print the result of the reduce operation