I spent the weekend playing this match-up. During Narukami's rushdown, there are a few parts where you can just use her Getsu-Ei to get out. And because Izanagi likes to stay out forever, it's very easy for Mitsuru to persona break Narukami which will really limit his options. If, for whatever reason, Narukami throws out a fireball, just j.D into a combo.
Personally, if I see Narukami with bar, I get worried because of his full-screen super. I'm still learning the game, so not fully sure how to get in with Mitsuru without eating a super.