The effort has not been without hiccups: Michael Roberson, a Republican who is the Senate minority leader, was infuriated when he learned the group had gone to his home to urge his wife to support the guns bill. A flier distributed by the coalition asked voters to call the senator to ask “why he wants to make it easy for convicted felons, domestic abusers, and the dangerously mentally ill to buy guns.”

But the broader bill, with background checks, passed the State Senate late in May, and the measure now has enough support to pass the Assembly. This weekend, in the final days of the Nevada Legislature’s session, Mr. Bloomberg’s team is concentrating on winning over a handful of Assembly Republicans, hoping a show of bipartisan support would prompt Gov. Brian Sandoval, a Republican, to reconsider a threatened veto. The coalition is also airing a television ad urging Nevadans to ask Mr. Sandoval to sign the legislation.

At one point, 11 lobbyists for the coalition were squaring off with one lobbyist for the National Rifle Association at the Nevada capital. But Billy Vassiliadis, who runs a prominent lobbying firm, said he quickly dropped out of the effort after deciding that Mr. Bloomberg’s battle was unwinnable.

“Taking part in a production, rather than actual work, was not in our interest,” Mr. Vassiliadis said. He said Mr. Bloomberg needed to spend more time educating the public about the benefits of background checks.

But Mr. Griffin, 40, has stuck with Mr. Bloomberg, whom he previously knew only from watching the MSNBC show “Morning Joe.” He and his team have become something of a spectacle in the halls of the Legislature, mocked by some as puppets of a billionaire, admired by others as symbols of smart campaigning.

Mr. Griffin has advised Mayors Against Illegal Guns on advertisements, gathered cellphone numbers of lawmakers to pass along to the coalition, even noting that one lawmaker was an avid country music fan and might benefit from a call from a celebrity in the mayor’s orbit.