Salute, I'm a saxophonist and I've actually invented a solfège system a couple of years ago for personal usage which can deal with quarter tones.
The basic principle is simple, keep the natural Italian note names, no matter you're using Si or Ti for the 7th note of the diatonic scale. Then, if the accidental is sharp, change the vowel to "u" (like "u" in tuna) and if it's double sharp than "ü" (of German language), if it's flat change the vowel to "ə" (like "a" in tuna)and if it's double sharp then "ɚ" (r-colored ə, like "er" of brother). For quarter tones if it's one quarter, add "k" as suffix and if it's three quarter than add "t" as suffix. For example, "Dut" refers to Do three quarter tones sharp (3/4 sharp), "Lək" refers to La one quarter tone flat (1/4 flat).
Since they're based on original note names, there is no problem using it as movable-Do. Also for all 63 syllables, no exceptions of the rules and no conflict syllables.
Here is an analysis-introduction video I made for further story:
The video is about 20 minutes long, if you don't have time here is the article version
. Hope it could give you some thoughts!