I don't exactly get what the problem is. Her concerns about being seen as Chiru's caretaker don't have anything to do with Suzu and Chiru being together.
If I were to guess the issue is the emotional significance of Suzu being just about the first emotionally significant relationship she's developed "privately", that is by herself outside the context of Chiru and her family. The usual teenage chafing for independence may also be involved. (Also gay crush given what kind of series this is but anyway.)
Basically she wants to keep the matter as "something of her very own" and starts bristling at Chiru's well-meaning butting in on her private business, not particularly helped by the onset of jealous anxieties she has little experience in coping with.