“We don’t give up,” said John Feinblatt, who oversees Mayors Against Illegal Guns and serves as Mr. Bloomberg’s chief policy adviser. “We’re here for the long haul.”

The coalition would not offer details on how much it is spending on state efforts, and Mr. Griffin, who was approached a few weeks ago by an emissary for the group, declined to say how much his firm was being paid. The coalition has spent tens of thousands of dollars on television ads in Nevada — a relatively inexpensive media market…

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.