There is an idea of nomenclatures, disseminated by Rick Beato, that generalizes the names of the triads. In this case, 1-♯4-5 would be a "lydian triad" (as seen in this video at 2:42 Why Triads Are: The Building Blocks of Harmony). A "phrygian triad" would be 1-♭2-5. These 2 triads along with major, minor, sus2 and sus4 form all the possible combination with a perfect fifth. Then the same process can occur with a diminished fifth and an augmented fifth (which has 2 combinations that are simply inversions of minor and major triads), generating names and symbols like C♭5, Csus4(♭5), C locrian (1-♭2-♭5), etc...
Your procedure is the correct way to generate suspended chords inside the scale: raise the middle note one degree using the notes of the scale (or lower it). You're also right about the 4th. A sus4 chord has a perfect fourth by definition. I'd like to call it an "augmented sus4" too, but unfortunately that's not an established name, although logical. If you really, really want to communicate just these 3 notes in a written scenario, maybe you could use the horrible symbol A♭add♯4no3 (or A♭add♯4omit3). If the 3rd is harmless to you and is even played by some instrument, you could say it's an A♭add♯4. If you have reasons to believe there's a "hidden" root somewhere else other than the A♭, it could become an E♭Maj7sus4 like Dom mentioned, or even a D locrian (a pronounced bass note D is an easy way to make that happen). But if it functions like an A♭ with a different "color/flavor" there's no need for that. If it's an informal situation I'd call it a "lydian triad" and try to make these names more well established, because they are useful in communication. About the symbol, it's a logical way for me to call it A♭sus(♯4) and I would, but that use is also at your own risk, since I've never seen it written.