Just take lots of pictures showing all of his unique markings, lots of pictures with you and him together, keep a file of all of your coggins papers, health certificates, any other vet work you have done. Being able to show a timeline of ownership will be more helpful than being registered. ETA: A record of your board payments will also help, since you said you have no vet paperwork. How do you not even have a coggins on him though??
Being "registered" really has no effect on whether a horse belongs to one person or another. There are lots of horses who are registered to a person from years ago, but the horse has been sold on without the papers being passed on.