I'm pretty sure he did want to get sued, yes. He knew Billy is litigious, and his behavior only makes sense if he was trying to bait him into a lawsuit. He thought he could bait Billy into suing him, make a million videos shitting on him, humiliate Billy in court, and then make videos about winning in court. He's an arrogant douche who's not half as smart as he thinks he is, and he was confident he could beat Billy with no effort.
I do not think Karl ever seriously considered the possibility of Billy Mitchell winning the lawsuit. He mentioned it one time in passing as a joke, and clearly didn't think there was any real chance he'd lose. Which, considering the subject matter and his defense, is CRAZY to think about. How can there not be a single moment of clarity where you read over the documents and go "wait, I'm fucking retarded. My defense is completely batshit insane. I should fix this shit before it goes any further."?