The short answer is "No," there isn't a standard way to refer notes with accidentals, using letter names, and taking up only one syllable—in English. The Germans, though, are all over it, adding "es" the letter for flats and "is" for sharps. Of course, it works best when you actually pronounce the letters themselves as they're pronounced in German; if your A is a Fonzie-like "Ayyy," it's harder to distinguish "Ais" from "As" than if it's an A as in "father."
But if you're inventing a convention for your own personal habit, you can pretty much do whatever works for you. Surely you can blurt out "geesharp" or "beeflat" on all but the shortest notes. Or for that matter, I would be inclined to leave the sharps and flats unspoken and simply sing "E" for E flat, for instance, and just understand that it's flat. Especially when they're part of the key signature; if I'm singing in a key with lots of sharps and flats, it seems unnecessary to be constantly saying them. I take a similar approach myself with chord qualities: If I'm thinking through a chord progression, I'm likely to sing to myself "C, F, D, G." The D chord is minor, but I don't bother saying "D minor"; of course the ii chord is a minor triad. On the other hand, if it is in fact major (as the V of V), I'm likely to mumble "C, F, D major, G."
(Meanwhile, I do encourage you to explore the benefits of moveable-do sight-singing, which is less about absolute pitch designation and more about tonal function. Although perhaps "confusing," it serves a totally different purpose, and a useful one. If you're learning "Rudolph the Red-Nosed Reindeer," then "G A G E C A G" tells you something about the physicality of the absolute pitches in your range, but "sol la sol mi do la sol" tells you about the melody, no matter what key it's in.)