I would guess that it was an after-the-fact constructed memory to justify why he followed and then had Julia Postal call the cops on a guy who he didn't see do anything wrong.
You may be right.... Perhaps Brewer was simply playing Dick Tracy that day....... But here's what bothers me.... He had just bought a New Ford Fairlane and he had it parked in front of the shoe store. He told Griggs hat he really didn't want to be there at the store that day, because he wanted to go riding in his new car. But his associate called in and said the baby was sick so he wouldn't be at the store that day, which forced Brewer to be there. Brewer could easily have hung an "out to lunch" sign on the door of the shop and took a lunchtime drive in his new car.....
There's something that don't quite mesh with Johnny Brewer tale..... And I doubt that anybody will argue that his tale is full of inconsistencies