Alright - this is really, really aggravating.

By default, using the dog race as a player results in a really messed up third-person camera view that traps you under the dog's chest, where the dog has also shoved any equipped weapons. I followed a tutorial to add a camera to the dog race, which works for most people including myself, but a small handful of users continue to report the issue. Assuming it was a mod conflict with another mod also doing something to dog skeletons, I created a new race, which works for me and removes any potential conflict...

but the issue continues to persist and I am going up the wall trying to solve it! What could be causing this?

I don't think just making a copy of the race is enough. I've been working with the dog myself and created a new dog race and by default, it will still point to the vanilla skeleton. Did you make a copy of the skeleton.nif and update the skeleton setting (in the Race dialog) to your copy?

Yes, I did.