I know Mongolian, and I can read and write Mongolian ancient script, Монгол Босоо Бичиг, and it uses a very similar vowel harmony structure. Mongolian has been seriously bungled by Cyrillic, reducing a natural set of vowels to something impossible to represent in Cyrillic.

I think the ancient Mongolian script possibly worked well because you could write something in a standard way, and the same script worked for any dialect that would read it.

Perhaps modeling your alphabet after the mechanics of a "vowel harmony structure" with a modern facade like Latin characters might work, dunno, I don't know Казак.