Yes, most of us are fond of pets, but realise that animals are animals that provide company and affection for their owners, but should be respected as the beings they are.
What this boils down to, is that very few dogs are comfortable at large gatherings, so that is one very good reason for not taking a dog to a wedding.
Another, even better one, is that you have been asked not to bring the dog. Unless you want to start a row with your DIL, obviously you cannot take the dog, and to be quite honest you are the one being unreasonable here.
You say you could prevent the dog jumping up on the bride's beautiful dress - are we talking about taking a dog to church? The only dogs I have ever seen in church were guide dogs, unless we are talking about a St. Hubert's Mass where hunting dogs are obviously brought on that one occasion to mass.
Presumably, you have time to either find a neighbour who can and will look after the dog for the day, or find a kennel.
If your son and daughter-in-law are intending to have a family, you may as well face the fact that the dog won't be welcome at the christenings or probably around the children.
Your daughter-in-law has just as much right not to like dogs, as you have to be fond of them.