> I'm curious, how much effect do lobbyists have on the FCC, since they aren't exactly elected officials?

The members of the FCC are all political appointees of elected officials, and, like most regulators, two of their primary future employment prospects after their term ends are (1) further political appointments, and (2) employment, often as lobbyists, in the industry regulated by the regulatory body they currently serve on.

So, while they may not be amenable directly to the same kind of campaign-support influence that elected officials are, there are good reason to think that lobbyists can influence them in ways beyond the persuasiveness of their substantive policy arguments.

The less ethical approach would be to insinuate to Mr. Wheeler that he'll be blackballed when he gets out of office.

The more ethical approach would be to lobby Congress to pass laws that override the FCC's decision. Republicans don't have a veto-proof majority, but that's never stopped anyone before - tricks with riders unrelated to the main bill can effectively neuter the veto power.