I don't think that it would really make that much of a difference. After going thru the lawn mower, it is going to be quite chewed up, and pretty well mixed into the clippings.
I believe that was included in the things not to mulch, because of its high content of nitrogen. If you were to get a clump of that in your garden it may over fertilize the local area surrounding it, and burn out any
plants.
The good thing about dog feces is it breaks down quite rapidly. If you are running 2 compost piles, one currently being made and one older that you are using. It should not make much of a difference. Just don't forget to keep turning the piles, that will ensure that everything is mixed well.
Mike