Technically, if you are involved in a traffic accident, you are required to stay on scene and give your information according to 32-10-2. With Barron being on scene, and being the owner of the vehicle, he is obligated to give his information to the LEO.
Furthermore, the 5th amendment does not apply to Barron's arrest. The 5th amendment simply states Barron is not compelled to be a witness against himself in a criminal case. Barron was arrested for not being smart.