Common sense says such a thing should not be allowed, but what does canon law say? Ed Peters says that canon law doesn’t explicitly forbid animals in the sanctuary during Mass, but that’s probably because there are an infinite number of things that shouldn’t happen in the sanctuary during Mass and you couldn’t possibly construct a law explicitly banning all of them. This is why Holy Mother Church relies on good sense and decorum. Unfortunately, those seem in short supply these days.
So, I guess we need a new law: No animals in the sanctuary, ever, and no animals (except certified assistance dogs) in a church. There, now I won’t be tempted to bring variously my two dogs, four parakeets, a cockatiel, salamander, newt, corn snake, or rabbit to church, and my fellow worshippers will just have to seek elsewhere for “the sense of calm and peace” I experience around my pets. Though I still think observing common sense would save us all a lot of trouble.
There’s another alternative as well: the chaplain’s bishop could tell him to stop being a maroon and stop bringing his dogs to Mass.