A Cabinet council chaired by Attorney General Edwin Meese III recommended yesterday that President Reagan support a bill to exempt state and local governments from having to pay an estimated$2 billion to $4 billion a year in overtime wages to police, firefighters and other employes.

The Supreme Court ruled last February that state and local governments must adhere to the Fair Labor Standards Act. As a result, they must pay overtime instead of providing compensatory time off to an estimated 7 million workers previously exempted from coverage by the law.

The decision prompted an outcry from local governments, and a denunciation from Meese, who told the American Bar Association last July that the ruling "undermines the stability" of state and local governments.

The National League of Cities and other organizations said the court's 5-to-4 decision in Garcia v. San Antonio Metropolian Transit Authority would impose a severe financial hardship on big-city public safety operations and on small towns, which would have to begin paying minimum wages to volunteer employes, school crossing guards and others.

The Labor Department announced recently that it would begin enforcing the ruling in October. It said that it would require police to be paid overtime if they work 171 hours in a 28-day period, and firefighters if they work more than 212 hours.

Los Angeles officials have said the ruling would force them to spend $100 million a year, and some municipal officials have said that it would force them to lay off employes and reduce services. Locally, the District government estimated that it would have to pay up to $15 million a year to comply.

The case originated with Joseph Garcia, a San Antonio bus driver who challenged the city's practice of paying overtime only when drivers worked on their days off and holidays. The American Federation of State, County, and Municipal Employees, which supported Garcia, said yesterday it would vigorously oppose efforts to overturn the ruling.

Republican Sens. Pete Wilson of California and Don Nickles of Oklahoma have backed a bill that would overturn the Supreme Court's ruling. But congressional aides say the measure will be strongly opposed by organized labor and will have little chance of passing the Democratic-controlled House.