I had a mountain cottontail before the patch that changed this coat color into a color changer. After reading the news that they have given us back the shiny coat, I pulled the one I had out. To my surprise, it had the new shiny coat. I dismissed it and re-summoned it, and it was a different color. Apparently, the color changer now randomly changes color among the various colors that this pet is able to come in.
For comparison, here is a composite picture showing my shiny mountain cottontail in the "new" shiny coat, ordinary white, and an older shot from before Patch 7.0.
So, it appears that even if you have not caught a replacement yet, you can still show off your shiny pet (you just need to be willing to summon/dismiss it until it is the color you want it to be).