With the San Francisco Municipal Transportation Agency’s parking and traffic management becoming a bigger political issue, plans are being revved up for a City Charter amendment that would hand those jobs to a new Department of Livable Streets.

At least that’s the working name among backers of the proposed Charter amendment, which would leave the MTA staff with oversight of Muni — but not much else.

The MTA board would still hear all parking and traffic matters, but the Board of Supervisors would have the final say over parking rules, stop signs and the like.

“The buck stops with the Board of Supervisors,” said Supervisor Ahsha Safaí, one of the initiative’s sponsors. “I don’t want to be held accountable for something I have absolutely no control over.”

Safaí cited his frustration over the MTA’s decision to reject a two-year effort by his Excelsior constituents to get a four-way stop sign at the corner of Avalon Avenue and Edinburgh Street — where a pedestrian was later struck and injured.

Supervisor Aaron Peskin, who is co-sponsoring the ballot move, said the final straw for him was hearing that Mayor Ed Lee, with support from the MTA, was negotiating with ride-hailing giants to turn parking spaces into designated pickup stops for Uber and Lyft.

The mayor’s office says the pickup stops are to make streets safer. But Peskin sees it as a giveaway to the mayor’s friends, especially tech investor Ron Conway — who Peskin has heard is tied financially to Lyft. But a source close to Conway told us he has no investments in that or any other ride-hailing company.

The Chamber of Commerce, business think tank SPUR and the San Francisco Bicycle Coalition are lining up in opposition to the proposed Charter amendment.

“We all agree there are improvements that can made ... but a Charter amendment coming out of the blue isn’t a solution,” said chamber Vice President Jim Lazarus.

Safaí and Peskin need four more supervisors to sign onto the Charter amendment to get it on the June 5 ballot. They’re confident they’ll get there.

San Francisco Chronicle columnists Phillip Matier and Andrew Ross appear Sundays, Mondays and Wednesdays. Matier can be seen on the KPIX TV morning and evening news. He can also be heard on KCBS radio Monday through Friday at 7:50 a.m. and 5:50 p.m. Got a tip? Call (415) 777-8815, or email matierandross@sfchronicle.com. Twitter: @matierandross