I've actually been tinkering with my own dog-headed race inspired, at least partially, by the cynocephalic legends, and I also gave them extra hit dice regeneration. I'm curious where you got that idea - I'm sure I had a reason for it, but I can't remember what.

On that topic, I think you're making the right call with tying the hit die to the first class level. It makes the race useful no matter what class you pick, but it avoids making this race feel like a blatantly better pick than others for a given class.