Short answer is "yes," and I believe he did as part of the initial filing ask for a preliminary injunction.
Another aspect of Nunes suit is vs. Liz Mair, a Scott Walker (and ostensibly Ryan and the rest of the never-Trump wing of the GOP).
This isn't just about who is going to dominate the internet, it is also a fairly direct attempted hit against a set of political opponents.
The demand for prelim injunction is narrow. It calls for suspending the named anti-Nunes accounts, and that's about all. @Liz, @Nunes-Mom, @Nunes-cow … as well as finding and disclosing the entities behind the "mom" and "cow" accounts.
The lawsuit alleges shadow banning of Nunes personally. The whole suit is personal to him as sole plaintiff. Ergo, the demand for injunction aims to expose and cut off particular detractors.