Ok, I've concluded that the ring absolutely has to go through.
I'm playing with tenex so it's a lot more for forgiving that whatever rope is in your original pic, Nick.
I did the small ring side first, and put the large ring on the other end leaving a long tail. I put the small ring through the braid by pulling strands to create a large opening for the ring. When I was done I pulled them back toward the tail.
Right track?
With a loose braid like tenex it wasn't bad. With whatever rope yours is it would take a bit of time to get everything back nice and purdy.