Could this be "The Anytime RingsThe Anytime Rings" (1963) by Robert Faraday? This involves some children traveling in time with a pet cynognathus. I don't know if the other details match up though.
GordonD contributes the following description:
The dinosaur was an exhibit in a carnival sideshow and it was the owner who had the time-travelling rings, which is how he got the animal in the first place.
The dinosaur was injured in a fire and the boys took it back to prehistoric times which presumably led to the adventure described above.
The only other thing I remember is that it was a Cynognathus nicknamed 'Nog' which also fits the description.