I'm not sure if these are the best examples, but I'm trying to draw a parallel between the difference in enharmonic spelling of something like a German augmented sixth chord and a dominant seventh chord, where the two enharmonic spelling reflect two different harmonic functions, and this case of an enharmonically respelled major triad.
The first phrase is just to set a key of
C minor. The second phrase is where the chord in question sounds like a Neapolitan chord so we want to spell it sensibly with an
A♭ for a half step above tonic
G. The third phrase spells the chord with a
G♯ as part of a chromatically ascending line, labelled
p.c. for passing chord, where the dominant/tonic relationship is the ending on a
In the second example the
N6 Neapolitan chord is a well known chord type and is considered a type of altered subdominant, a
iio diminished chord with the root lowered. If the linear movements coincide with those particular functions, it makes sense to enharmonically spell things to make the notation explicitly read as such.
In the third example the chord in question could by an
A♭ triad in key
Cm, which in terms of function is
VI, a fairly "weak" modal chord, but the phrase doesn't continue in
Cm. The chord could also be labelled
♭VII in key
B♭, but that again would be a weak modal type chord. In this case I think the chord truly is ambiguous in terms of harmonic function, and that ambiguity provides sensible leeway to spell the chord one way or another. One of those options is to spell it to highlight the chromatically ascending line. And that choice then give us a minor triad with the chordal fifth raised. Technically, because of its spelling, it is not a standard triad type of either major, minor, diminished, or augmented. Functionally it's a harmonic non-entity. (It doesn't fulfill predominant-dominant-tonic harmonic function so there isn't much point in giving it a Roman numeral analysis label.) It's just the result of contrapuntal movement.
You might also make a comparison to the 'classical' harmonic label French augmented sixth chord with the jazz label for the same chord. In jazz it's called a dominant seventh flat five chord (or depending on context it could be called a tritone substitution chord, but that's another topic.) Notice of the jazz label is a literal description of the alteration of a standard 'classical' harmony chord type? That's basically the same as how you presented your question. You, in essence, called a first inversion
A♭ major triad a
C minor sharp five chord. If jazz theory can use such process oriented labels, why can't you?
One final thought. Just because something doesn't have a conventional label does not mean it is musically meaningless. Think of a conventional label as a convenience. Something that make a musical action easy to talk about. Absent a convenient label you can still see and hear and describe how something is working. Things like root progression,
FA-MI dominant function, harmonic sequence, melodic/contrapuntal motion, non-chord tone identification, and resolution of dissonance are some of the big concepts that come up in analysis.