I'm not sure he's looking for a lawsuit, just attention. I am pretty sure Russ thinks causing all this will eventually get Taylor to talk to him, or get a huge media outlet to talk to him so that he can use what happened to him to get something. Like a woman who takes a minor inconvenience at a retail outlet and tries to get a free $1000 gift card. Might be 15 min of fame, money, pussy, something. Not sure what's in his weird little gourdhead.
However we have seen he's afraid of being sued himself. When Ariana's team gave him the C&D, he caved but boasted later like he pulled a sneaky on them. Notice when the family of the girl who commited suicide asked him to not have her in his book, he ignored them. Earlier he sued Farrah Abraham and she slapped a countersuit, he immediately backed down, lied about it, and then just sorta let it go. The restraining order(s?) got him really mad but I think he still abided.
I think he's slowly learning the legal system is not his personal bully or a tool to legally rape women. He basically let the Taylor lawsuit go and went straight to his dumb little song. What's funny is that like his love life, Russ has learned you can just pay your way to the end goal. He's got a professionally produced song, the photographs, someone putting his song on music platforms, etc. He thinks this makes him some kinda big shot producer but they're all just rolling their eyes and taking the checks while they watch a crazy person make art. It's kinda like the making of The Room without the possible cult status.