Do these diminished chords fit in a diatonic scale well, even though they're chromatic?
This is not a yes/no question, but rather, the answer is "It depends". If you want your melody to remain purely diatonic, you would probably want to focus on notes that are in the key of C, and that also are members of the chord at hand.
For instance, in CM7, you have all four notes that would work: C E G B. Then in Dbm7, only Cb would fit (it is the same frequency as B; Db Fb Ab do not reside in the C scale).
You could continue to use this method to build a melody that is strictly in C, and in fact, all your chords have at least one note that is in C. Although you could try some notes outside of C as well, if you're interested in experimenting.
Now, whether they fit "well" is a more complicated question. You'll need to go one chord-melody combination at a time, and possibly need to go back and rewrite until you have a satisfying melody. There is a lot of study necessary to work out such a complicated chord progression.