Yes, this strikes me as an example that willfully invites multiple correct answers.
Your three solutions are correct, but there is one other possibility if we change mode. In addition to your first solution in F♯ major, you could also have one in F♯ minor, which would involve G♯–B–D. This chord is almost always found in first inversion, which would be correct with the subdominant in the bass.
The two other keys don't work in minor: first because D♯ major isn't a real key (it requires two double sharps in its key signature!), and because B minor requires a G♮.
I wondered if perhaps Piston was slyly hiding in a doubling rule here; some books/teachers say that first-inversion chords can't have this or that doubled, but he has made no such claims in this chapter (p. 17: "Doubling of the root is most usual, but the third or the fifth may on occasion be doubled instead."). As such, any of these four options would be correct.
Furthermore, there are multiple realizations for each of these correct answers: as one example, the soprano G♯ can either be on the second line of the staff or above the top line, both of which fall into Piston's suggested soprano range on p. 16 (even if the high G♯ is really pushing it).
Edit: After considering Peter and Matt's comments regarding the use of "mode mixture" chords, I decided to look a bit more closely at the book. At this point the text has only discussed triads in the major mode; triads in minor wait another two chapters. As such, any minor examples were likely not intended answers, and they should probably instead be limited to the B and F♯ major examples in the original question.