Per an SI article, Len Barker did not allow three balls to anyone in his 1981 perfect game.
He struck out 11 batters—all swinging—and didn't go to a single three-ball count. Only eight Toronto batters got as much as a two-ball count.
Your fact finding of it taking place in Anaheim is correct based on announcers. I think it is safe to assume based on uniform colors that the team on the far side is the Ducks. They featured a light color top home jersey at that time.
Placing an identification on the opposing team is more difficult given video quality. Based on this schedule, it would ...