Hmm, interesting question. I assume you're using uppercase and lowercase Roman numerals to distinguish chord quality? If so, I see two options.
Accidentals and Roman Numerals
D major, as you've correctly said, is best shown as V/V and not II. But F♯ wouldn't be V/vii°, because that doesn't exist. But it would be silly to write it as V/V/iii, which sometimes does happen. Instead, this suggestion would be to write F♯ as ♯IV. If this F♯ chord were diminished, however, you could show it as vii°/V (not ♯iv°). D♭ would be ♭II, D♭ minor would be ♭ii, and so on.
This approach follows in the tradition of Heinrich Schenker; you can find his use of these Roman numerals in scholarly discussions of "secondary" and "double" mixture.
But that leads me to wonder how helpful that would be for high school students. (Maybe this is unfair of me to say, since I don't know the class or the students.) I wonder if throwing in all possible chords is that helpful? Are you really intending to use chords like ♯vi and ♭iii? Which leads me to my second recommendation:
Stick With Chords That Actually Occur in Music
By doing this, you could get rid of the nonsensical chords like ♭iii that either don't really happen, or aren't encountered until much later in one's music theory training. By doing this, it seems you could stick to the following chords in a major key:
- I, ii, iii, IV, V, vi, vii° (and any sevenths you want)
- Applied chords to ii, iii, IV, V, and vi (with sevenths as you like)
- Modal mixture of ♭III, ♭VI, and ♭VII.
- (You decide whether to include ii°, iv, and vii°7.)
- The Neapolitan ♭II.
This latter option, in my opinion, seems to mimic real musical practice, and therefore the actual music theory that they'll be using.