After the battle I promptly sought out a wild Mountain Cottontail on the PTR and found one as a secondary pet. It took a few restarts, but I managed to get it spawn in with its new shiny appearance.
It seems the special version of the cottontail will be back in 7.1! (see above screenshot for glossy skin on the left and normal white on the right)
Unfortunately the updated look doesn't seem to be retroactive. The existing Mountain Cottontail in my Pet Journal on the PTR (was originally shiny) did not have the new skin. Instead, it remained a color-changer so it had the chance to be spawn in with the glossy appearance, but it wasn't guaranteed. Capturing a new cottontail that started out with the shiny skin guaranteed that it retained the look.
It's a little inconvenient that we'll have to capture new Mountain Cottontails in 7.1, but on the bright side we won't have to travel to Redridge to find these wild bunnies. They also appear in Highmountan in Legion, hooray!
I'm ecstatic that this special wild pet will be making a come back. Big THANK YOU to Blizzard for maintaining these small details that mean so much to some collectors. It's the little things that make the game that much more fun and amazing. ❤