Sure it is OK to include your extended family (albeit a little hairy) family member in the ceremony. Remember, that the lovable pooch may never had to walk down an isle in front of a bunch of people before. How they react can be completely the opposite as you would want. They can be terrified, or drink up the attention.
If they are going to play the part of ring bearer, in my experience, it’s best to have the “real” rings with the Best Man. Nothing can break up a ceremony faster than having to chase your animal down to get the rings for your ceremony. You’ll also want to make sure you have someone designated, before the ceremony, to take care of your pet during the ceremony.
(These same tips also apply to very young children.)