Should I do this?

My ex stole my dog and I recently filed for small claims so I can get my baby back. I have vet records in my name, receipts of things I purchased for the dog, a copy of his lease stating that he can’t have that specific breed at the house, messages between he and I showing that I didn’t agree to letting him have the dog, and messages from the person I got him from indicating that I was the one who paid for him. My question is should I go ahead and get him registered in my name since I have vet records even though I don’t have him back yet? It would be further proof that he belongs to me