I only realised this chapter that most of the siblings have "moon" (tsuki) in their name, at least on the face of it. The exception being Tsumiki.
Tsumiki, Itsuki, Natsuki, Kizuki. Kizuki not being Kitsuki is probably a consonant change because of grammar rules. But I don’t speak the language, so this is effective just guessing.
I think you're right on the tsuki thing, Tsumiki just has the 'mi' in between the tsu and ki. Though interestingly in the first weekly episode in the magazine, Tsumiki's name was originally given as the kanji 積木 which means 'stacking/volume' and 'tree/wood'. After that single time those kanji were never seen again, it just went to つみき (tsu mi ki). Maybe so she could do the tsuki thing with the siblings!
And yeah, often (but not always) when a word gets added to another word, the first syllable in the added word gets 'promoted' to a harder sound. Like しま ('shima', island) becomes じま ('jima') in the names of a lot of islands like Miyajima. Though there are other cases where it doesn't, like Enoshima. Critically, syllables are always promoted to the dakuten (two little marks in the upper right) version. か (ka) becomes が (ga), は (ha) becomes ば (ba), etc. In this case the つき (tsuki) becomes づき (dzuki). Dzu is often romanized to just zu (even though there's already ず for zu) just because the D is very soft and short, and if you see 'Kidzuki' the temptation is to pronounce it 'Kid-zoo-key' when it's really 'Key-dzoo-key'.
There's no rule on when something gets promoted like this, native Japanese speakers just seem to 'know' (ha gets promoted to ba a lot), and even they often disagree. I looked at a bunch of island names and could see no rhyme or reason between -shima or -jima.