Ms Babbit was fatally shot on the day of the attack while trying to climb through a window into the Capitol's interior chambers.
The woman was 35-years-old at the time she was killed, and wore a Trump campaign flag draped around her during the attack on the Capitol.
The Capitol police officer who killed Ms Babbit has not been charged with a crime and has never been publicly identified.
However, Terry Roberts, the lawyer representing Ms Babbit's family, told Zenger news that he knows the name of the officer.
"A rookie police officer would not have shot this woman," Mr Roberts said. "If she committed any crime by going through the window and into the Speaker's Lobby, it would have been trespassing. Some misdemeanor crime. All a rookie cop would have done is arrest her."
He claimed the officers - who were facing down a crowd of thousands of Trump supporters trying to breach the building - could have simply arrested her.
"And he has plenty of other officers there to assist with arrest," the lawyer said. "You had officers on Ashli's side of the door in riot gear and holding submachine guns. And on the other side of the door you have another uniformed officer 6 or 8 feet away. Whose life is he saving by shooting her? ... She's not brandishing a weapon. She's on the window ledge. And there's no reason to think she's armed."
Mr Roberts said the family was seeking $12m in damages and believes the officers involved will be held personally liable. He does not believe the officers will be covered by qualified immunity protections that generally protect police during shootings that occur during law enforcement actions.
Ms Babbit was the only rioter shot during the Capitol attack. Three other Trump supporters died of medical issues during the insurrection.
One Capitol police officer, Brian Sicknick, died after being struck on the head by a Trump supporter, though later medical reports concluded he died of "natural causes."
Another officer committed suicide shortly after the insurrection.
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