Find locations where you will hurt Verizon customers, and cut the cables. Do so publicly. Precondition repair on upgrades of Verizon's network as you direct. If Verizon doesn't want network neutrality, then punish their customers.
I wish it was that simple, but I'm on board with the general idea. I wouldn't publicly cut the cables. That's too extreme. I would, however, like to see Level 3 turn the tables and publicly (as noisily as possible) accuse Verizon of using up all their bandwidth and that if Verizon doesn't help them pay for the costs of upgrading, their customers just won't be able to watch Netflix anymore. "I don't see why Verizon is taking issue with this. It's standard practice. People download a lot of video from us. Remember folks, we're not downloading from them! Verizon's users are downloading from us! They're the ones using all the traffic!" (I know that's a completely inaccurate and misleading explanation of the situation - an outright lie, if you will - but that's the point.)
If Verizon customer's thought "Who's Level 3? Netflix is paying them so I can watch Netflix, and now they want to charge Verizon money so I can watch Netflix? So basically I'm getting charged twice to watch Netflix? This will not stand!"
Then when everybody has turned their heads in their direction, Level 3 would say, "Just kidding! Here's what's really going on..." and tell people a simplified version of what they said in TFA. Then maybe people would start to care about net neutrality. (If you were to replace Level 3 with Netflix themselves, it would oh, so much better!)
Of course, it's a good thing I'm not in charge of either company because I'd have just lost an unprecedented amount of business and ruined the company's reputation in the process, and unfortunately, the average person isn't going to respond a calm, well-reasoned, fact-based argument like TFA. Most people just aren't going to care as long as there's something playing on TV. When you get in the way of that, people will definitely notice, but anybody even remotely involved in such an affair will be ruined for life.