Gio, you're right that not all churches would allow a dog to be part of the wedding ceremony inside the church. Sometimes, though, it's possible to ask special permission.

On a sadder, note....a young man was killed in tragic circumstances here & the (anglican) church gave permission for his much-loved pet dog to sit with his family during the funeral service inside the church.

Also, when a police-dog handler was killed on duty here, the (catholic) church gave permission for his police-dog to be inside the church at the funeral.
The dog, with another handler, led the procession out of the church. Outside, over 30 other policedogs were lined up as a guard of honour.
Marie