I like this idea. If I understood correctly, Christopher would stay on for several episodes while the crew is working on a solution for returning him to his own time. This would totally work in a modern day series where successful series can pretty much plan out their episodes for each season. Ultimately there would be "a very special episode" where the solution would be found and Christopher would leave after some tearful goodbyes with the regulars and possibly a touching scene between him and Kirk or Spock.
Trouble is, I don't think TV series of the 1960s dealt with that sense of continuity that is so common today (where TV dramas and comedies have a structure that resembles more that of the soap opera). Each episode was made as a self-contained story (which is why you can jump around from season to season and not find yourself asking a lot of questions about who the new characters are and why are these two dating now because I thought they hated each other...etc...).