With reprimands for both sides, a federal judge refused Thursday to let San Francisco enforce an ordinance that would subject Airbnb and other short-term home rental companies to fines and criminal penalties if they let homeowners rent rooms without registering with the city.

After U.S. District Judge James Donato rejected most of the companies’ challenges to the ordinance last week, San Francisco’s Airbnb and a second major vacation-rental firm, HomeAway, came to Thursday’s hearing prepared to concede defeat in Donato’s court and take their case to a federal appeals court, while seeking a continued stay of enforcement. With the case seemingly resolved in Donato’s court, lawyers for the city wanted immediate enforcement.

The judge was unimpressed. “You have not done the heavy lifting,” he told the opposing lawyers. He ordered them to start negotiations promptly, supervised by a federal magistrate, on a workable system to register all short-term rentals with the city.

“This can be solved,” Donato said, setting a mid-December deadline. In the meantime, he maintained a bar on enforcement of penalties on the companies for booking unregistered rentals.

“We're pleased the court has ordered Airbnb and HomeAway to actually come to the table and work with the city on an automated system for enforcement,” said John Cote, a spokesman for City Attorney Dennis Herrera.

Airbnb, which had fought measures requiring it to police its hosts, suggested a willingness to cooperate this week, saying it could provide San Francisco with its local hosts’ names, addresses and guest stays as part of a mandatory registration system.

The company is working on a similar system in Chicago.

But lawyers for both Airbnb and HomeAway said Thursday they still intend to appeal Donato’s ruling last week that concluded San Francisco’s registration requirements would not violate their rights under a federal law that shields Internet hosts from liability for messages posted by others. Airbnb, which has a private-market valuation of $30 billion, has retained Donald Verrilli, formerly the Obama administration’s top Supreme Court litigator, to handle the appeal.

San Francisco supervisors, meanwhile, voted this week to limit short-term rentals of private homes to 60 days per year, down from the current 90 days. Mayor Ed Lee may veto that change, however.

The city has required property owners to register short-term rentals since February 2015, but only 1,700 of the estimated 8,000 to 10,000 hosts have done so. That gap led supervisors to enact the ordinance that would hold Airbnb and other companies liable for unregistered rentals by hosts using their platforms.

The companies contend the registration process is cumbersome, requiring multiple home visits. Airbnb has proposed instead to automatically register all hosts using its service, and to cut off their listings when they exceed the maximum rental periods.

Bob Egelko is a San Francisco Chronicle staff writer. Email: begelko@sfchronicle.com Twitter: @egelko