The police shooting of a 33-year-old man in a California Walmart over the weekend has led to intense backlash from civil rights activists, calls for protests and a Facebook video from the local police chief to “dispel some rumors” about the incident.

Police in San Leandro in the Bay Area shot Steven Taylor on Saturday afternoon after he wielded a baseball bat inside a local Walmart. A video shot by a bystander captured two officers pointing their weapons at Taylor holding a bat near the doors on the Walmart floor.

The footage appears to show one of the officers deploying a Taser after Taylor had dropped the bat on the floor and was lying on the ground. One witness is heard shouting, “Don’t shoot him no more!” Police said one of the officers hit Taylor with a bullet in the upper torso, and the officers tried to use their Tasers multiple times during the confrontation.

Lee Merritt, an attorney for Taylor’s family, said Taylor was going through a mental health crisis on Saturday afternoon, and that he has previously suffered from paranoid schizophrenia and bipolar depression. “He was shot after he had become completely helpless and no longer represented a threat,” Merritt told the Guardian on Monday.

Merritt said he wasn’t sure yet whether police shot Taylor with a Taser or bullet after he was already down, and that an autopsy was under way.

Merritt also alleged that the officers provided insufficient care once Taylor was shot. “Their job, according to standard operating procedures, was to get Mr Taylor help. He had been seriously wounded and was suffering from a mental health crisis. They had to treat him quickly. They did the opposite and exacerbated his injuries,” Merritt said.

Steven Taylor’s family is calling for charges against the officers who killed him. Photograph: Courtesy of Lee Merritt

The San Leandro police department said Taylor had not complied with officers’ commands to drop the bat and had walked toward police. At this point, one officer discharged his Taser “which was not effective”, according to the department. Then, police said, the officer fired his gun at Taylor, hitting him in the “front of his upper body”. Seconds later, another officer discharged his Taser at the man, according to the department. Taylor died at the scene.

Taylor’s family is calling for charges against the officers. Merritt, who represents families of those killed by police in federal litigation, said the officers should face homicide charges for targeting Taylor after the threat was “neutralized”. He said police should have de-escalated by clearing the Walmart, surrounding Taylor and trying to talk him down, instead of quickly using lethal force.

The San Leandro police chief, Jeff Tudor, said in an interview that the “pop” heard on the video after Taylor was already on the ground came from a Taser, and that it was too early to speculate whether that shot had hit Taylor or whether it was justified and in line with department policy. One officer was initially “trying to de-escalate the situation and grab the bat”, Tudor said, adding, “It’s very tragic.”

On Sunday, Tudor publicly acknowledged that the shooting had upset many. “Our community is hurting right now,” Tudor said in a Facebook video. “But protecting the sanctity of life is extremely important. I know there are a lot of questions and concerns.”

Few details have emerged about Taylor since he was killed. Merritt said Taylor had three children, including an 11-year-old, and that he leaves behind three siblings. “I hope they don’t see their father executed like that,” Merritt said.

He added that Taylor “was best known for trying to make people laugh”.

The fatal shooting happened just south of Oakland, in a region where residents for years have organized Black Lives Matter protests against police brutality and police shootings.

Last year, California adopted the strictest law in the US limiting when police can kill, dictating that law enforcement must “reasonably believe … deadly force is necessary to defend against an imminent threat of death or serious bodily injury”. Typically, courts across the US have long ruled that shootings are justified if officers claimed they feared for their lives and were acting in self defense, a bar that advocates have said was too low and allowed police to kill civilians with impunity, particularly unarmed black Americans.