Really you're talking about augmented triads, specifically, and yes, all augmented triad inversions create another augmented triad.
You already know which augmented triad you want, so this is more a question of accidental notation--this can come up in many other circumstances as well, especially if you're working in a neotonal context or are very far away from your key center.
If you're writing traditional harmony, the answer is probably going to be whichever notes fit your key signature the best. #2 and b1 (xD and nC in your key of C# minor) are rare, so the notation you're looking for is probably G# - B# - E. This is also really the same as finding one of the chords diatonic to your key signature. In C# harmonic minor, the only diatonic augmented triad is indeed rooted on b3 (E - G# - B#).
If this is not the case, then in order to answer the question properly, we'd have to know how you got to that chord and where you're going. i.e. how does the augmented triad function in the chord progression you've written. You would then choose an inversion that most clearly describes the root movement of the bass, or how you're planing something, or a weird deceptive cadence, or any number of other compositional techniques.
One last thing worth noting is that the first method I mentioned (look in the key signature when writing traditional harmony) is really exactly the same as the last one (do what makes logical sense): when you're writing traditional harmony, the logical progressions generally all fall on the diatonic notes, or not far from them in the case of secondary dominants.