Who exactly is the "opposition media" to you? Opposition to who? The White House and Republicans I am assuming... Unfortunately all media and news outlets are opposition these days as they ALMOST ALL have an opinion which then comes off as an agenda therefore they are opposition media to anyone with a differing opinion. Lack of reporting on a specific topic influences sheeple as much as over-reporting on it.
Democrats (and related media) will definitely latch on to and report on the Kavanaugh confirmation in order to fire up the left (especially women) to get to the voting booths, but they don't necessarily need to as they are already all fired up. This situation got so out of control and turned so political that both sides of the aisle tried to use it for gain and will continue to do so. Reporting in the media will be about whatever angle benefits their side's interests the most. Dangerous territory we are in...
You asked the question and then answered it too, although I don't think you intended it to be a rhetorical question
The opposition media for Trump and conservatives as a whole is every news source in this country sans Fox News. NBC, CBS, ABC, CNN, MSNBC, NYT, WP, Time, Late Night, Hollywood and even Silicon Valley. They spin every issue against Trump, everything, they're narrative is whatever Schumer and Pelosi say it is. They even share the "expression of the day". If one of them says "it's our country's darkest hour", they all do.
They refuse to to provide mitigating facts, there's zero balance in anything they do. They won't talk about all the accomplishments of this administration, they'll even give Obama credit for today's economic numbers by showing him take credit without any retort whatsoever. That's all part of the charade they're playing. They create false narratives and never retract anything when proven wrong. They even managed to make the accusations against Kavanaugh look credible, when it's some of the dumbest shit I've ever seen in the political arena.
We are in a dangerous place. Our free media, intended to be the protectors of democracy, is closer to being treasonous than they are objective. Something has to change, they can't keep dividing us, or maybe I should say "we can't keep letting them divide us".
As a nation, we have to start focusing on substantive policy issues and ignoring all the wedge issues they create to advance their own political beliefs.