The campaign by the Shaw family put some council members in a delicate position.
"We are best when we are one community," Reyes said. "What if a witness in this case was undocumented? What's going to motivate them to step up if they are looking over their shoulder at the police?" he said.
In addition to the Shaws, several anti-illegal-immigration activists spoke out against the order.
Councilwoman Jan Perry expressed concern at the overall tenor of public comments Tuesday, although she said there could be "something within the idea they have that they can expand on."
The Shaws wore T-shirts with "jamielslaw.com" printed on them. That site links to the campaign of Walter Moore, a community activist running for L.A. mayor who has criticized Special Order 40.
Last year, two groups sued the city over the order. One cited an obscure state code that appears to require local police to report the names of any illegal immigrant arrested on suspicion of drug sales or possession to immigration authorities. The other alleges that the policy violates state and federal laws that require local cooperation with immigration agents.
Sterling "Ernie" Norris, an attorney for Judicial Watch, one of the plaintiffs, said the Shaws' proposal to identify illegal immigrant gang members at the point of their arrest doesn't go far enough.
"Police aren't taking the next step, which is when they pull somebody over to immediately check into whether they are illegal through a thorough background check and through their investigation at the scene," Norris said in an interview. "Special Order 40 essentially told them to shut up and not to talk about illegal immigrants."
Several immigration rights advocates said they hope city officials will continue to back the order despite the emotional appeal of the Shaw family.
"In this country the law is that if your are a legal permanent resident or an undocumented immigrant, you face the additional possibility of deportation for having committed that crime," said Angelica Salas, executive director of the Coalition for Humane Immigration Rights of Los Angeles. "Both the LAPD and L.A. County sheriff already use immigration law to combat crime by questioning and identifying gang members and their immigration status."