The largest police union in New York City asked a judge on Monday to allow unvaccinated police officers to continue working, despite the city’s recently imposed vaccine mandate, which requires all municipal workers to have received at least one coronavirus vaccine dose by Nov. 1.

In a lawsuit filed in Staten Island, which is home to many police officers and has a vaccination rate that lags behind the citywide average, the Police Benevolent Association of New York said it opposed a vaccine mandate for police officers that does not allow the option of being tested weekly instead of being vaccinated.

The lawsuit also claimed that the mandate — which the mayor announced last week — does not contain sufficient protections for officers who might object to the vaccines because of religious beliefs. Mayor Bill de Blasio has said that the city will be “offering religious accommodation,” but that “valid religious exemptions” are rare.

While most lawsuits trying to stop government vaccine mandates in New York and elsewhere have failed to gain traction, some federal judges have appeared more sympathetic to suits that narrowly attack vaccine mandates for not accommodating religious beliefs.

Police unions across the country, from Chicago to Washington State, are urging members to resist Covid vaccine requirements — despite Covid being by far the most common cause of officer duty-related deaths this year and last, according to the Officer Down Memorial Page.

The New York police union’s lawsuit argues that the city did not give officers enough time to seek religious exemptions. Officers seeking exemptions are required to apply by Wednesday — one week after the mandate was announced — to avoid being placed on leave without pay.

As of last week, about 70 percent of employees of the New York Police Department had received at least one shot of a coronavirus vaccine. The P.B.A., which represents rank-and-file officers, has been generally supportive of an earlier policy that had allowed unvaccinated officers to test weekly for the virus. The lawsuit claims that “test-or-vax” rule was effective in protecting public safety.

The lawsuit was filed on a day when a large crowd of people — including many fire, police, and sanitation workers — marched in protest against the vaccine mandate. Walking across the Brooklyn Bridge to City Hall, some demonstrators carried large American flags and loudly chanted, “We Will Not Comply.”