Plot hole: The Bradys and Alice run a relay through the amusement park to get the sketches to Mike at the manager's office. They locate another person to pass the cylinder off to when they get too tired to run. But they all arrive together seconds after Carol gives the sketches to Mike. (00:19:15)
Plot hole: When Mike was at the meeting and pulled out the Yogi Bear poster, he said he could find the other cylinder, and was told by one of the other men that they had to catch a plane in 30 minutes. After that, he had to find Jan, and then Jan and Marcia had time to go on about 4 rides and then find the sketches and then start their run across the park. There is no way all of that could have been done in less than 30 minutes.
Answer: The dangling bracelet was used for dramatic purposes - to keep the audience on edge.