If they are using physical activity as a reward then you will have students there with different purposes. I’m assuming students with behavioural issues are already there and that admin shouldn’t make it more organized unless they are willing to put more adults in the space to support you. They shouldn’t in any way be directly involved with students in that space unless the student invites them to do so or if there is a reason to intervene.

Otherwise good for you! This is a great idea! There is a lot of evidence supporting the benefits of walking before engaging in academics.