Since the Gaku Joshi and the Kokin Joshi are physically so alike, it might come down to looking at your repertoire of taiko tunes that you might accompany, and seeing whether more of them have F# than G, then skipping over the note that the other songs have. As in, if you choose to own an F# and you play a song with a G, just don't strike the missing G, but choose a suitable harmonic to replace it.
I've just played the notes on a piano in order, to see which scale was more pleasing and I discovered that (at least to my Western ears)
C Eb F G Bb (Gaku Joshi) and
C D F G A (D Minor) sound pleasing but although the performance you linked sounds fine,
C Eb F F# Bb (Kokin Joshi) sounds a little off.