The footage shows Deborah and the unidentified man, thought to be in his twenties, shopping at a Festival Food store at around 5pm on that day.
The supermarket is around a mile and a half from the home in Kansas City, Missouri, where Lisa was abducted.
The PI would not identify who hired him but said it was not the child's family.
Police said he would not have access to investigative files. "He will have access to anything the general public has access to."a spokesman said.