Again, the short answer is: No. The same as you do not need to register with the LA, you don’t need proof or ID to prove your child is home educated for the Truancy Patrol, discounts at attractions or anything else.
Truancy patrols are just that. Patrols to look out for children who are truant from school. This does not apply to children who are home educated as they are not truant from school. If your child is home educated, all they need to do is say that they are indeed home educated and legally, the truancy patrol have to allow them on their way. They cannot detain them without reasonable cause.
If you would prefer something that your children can carry with them, that is your decision. You can print off your own card with the appropriate legislation printed on, and there are sites that you can download a template from for free. Your child’s name, photo or any identifying features do not need to be on it.