I think you'll have to wrestle Barry's owner for the right to walk your dog in the park.
I've checked the bye-laws, and it's in paragraph 4 on page 7 of "owning a dog in London - 3rd edition (2024)".
What's left is for someone to run a book and sell tickets.
Q1 : Can Barry's owner reliably recall Barry?
Q2: Is Barry's owner always going to be keeping an eye out for Owen?
Q3: Is there another park nearby?