Beckles and Martinez want city officials to update the job specifications for fire marshal and look into the staffing levels at the department's fire prevention division.
"Many of us on the City Council don't believe that we're living in a city where we can afford to have these two huge responsibilities be on the shoulder of one person," said Beckles.
The City Council has no power over such staffing decisions. Beckles said the proposal is aimed at convincing city officials to make changes, but she's aware that it carries no official weight.
"We can state very clearly that this is not something we are comfortable with," said Beckles, who is running for a seat in the state Assembly.
Mayor Tom Butt is opposed to the proposal. He criticized Beckles' measure in an emailed newsletter to constituents.
"She should learn to understand city government first before asking you to move her to state government," he wrote.
The mayor and Beckles recently got into an argument during a City Council meeting that led Butt to storm out of the proceedings.
Beckles and Martinez are members of the Richmond Progressive Alliance, a group that has been involved in strong disputes with the mayor in recent years.
Butt released a letter that City Manager Bill Lindsay wrote to members of the council, urging them to table the item.
"First, let me state that there are important underlying policy issues that are contained in the agenda item put forth by councilmembers Beckles and Martinez," wrote Lindsay. "Because these issues are important, it is most unfortunate that they have been framed as an unfounded personal attack on the fire chief."
Lindsay, who is stepping down later this year, noted that the City Council has already determined that the fire marshal job should be a stand-alone position and that the city's human resources department has begun the process of recruiting for it.
He also said the council could review staffing-level issues as it discusses the city's upcoming budget. The issue about the city's scrutiny of Chevron should be addressed separately, he said.
"A vote of 'no confidence' by the city council, relating to any city employee, undermines the proper administration of the personnel system of rules contemplated by city charter," wrote Lindsay.
Sheppard declined to comment.
He has said in the past that fire officials were concerned the sound of sirens during the Sims Metal Management fire would have prompted people to flee their homes when the department wanted them inside because of the toxic smoke.
Earlier this year there were three malfunctions at Chevron in a span of two months. The Fire Department normally does not investigate Level 1 incidents that do not prompt evacuations or shelter-in-place alerts.
Beckles said local fire officials rely too much on Chevron's assessment of flaring incidents and should scrutinize the refinery more.
"The fire chief/fire marshal is taking this company's word for it rather than investigating himself," said Beckles.
The Bay Area Air Quality Management District conducts probes into many refinery malfunctions. Sometimes the Contra Costa County hazardous materials program will require more information from Chevron after serious refinery problems. And other agencies, like California's Division of Occupational Safety and Health (Cal/OSHA), investigate serious refinery accidents.