Has Mitsuki even said she wants to chase the music industry in that way? It's been a while so maybe I've forgotten.
No. Thus far she's only been making music for herself and lately for Aya. She only decided to go to the Christmas party at Hime's house to meet her music producer father after being pushed by Aya, uncle Joe and Kanna, and she ended up ditching it anyway. It's very likely she'll eventually go that route since she did end up meeting Hime's father later and he noticed she's a diamond in the rough, but as of now she hasn't expressed the wish to go pro yet.