All the animals that are described as blooded have either become blooded
at Deismaar or have become blooded because they were involved in the
death of a scion.

I've seen no animal that has been invested or that has inherited the
bloodstrength of a parent.
Therefore the most simple solution is saying that the gods don't allow
investiture of animals or the passing of the bloodline to offspring.

But if you allow it you can ofcourse get animal regents.
All hail the holy bunny ruler of Talinie and it's temples.
(Great way for those non blooded regents to work, they just charm a
blooded bunny and use speak to animals and say they adminster the
holdings in name of the bunny

Jan Arnoldus