[Faber]

would it make more sense for me to organize them as movies or as Doctor Who specials under TV shows?

TV Shows. Some of these specials are only a few minutes in duration, so not sure what benefit there is to have them listed as movies.

Also be aware that the specials numbers listed at TVDB and TheMovieDB are different, so make sure of which scraper you are using and that you are referring to the correct site when checking numbering. 

Thanks!