Mutts do not belong in a house of worship, as they are known to take a piss/dump in the most disrespectful of ways.
What if the mutt bites somebody?
Or humps the bride’s leg?
Bad plan.
BUT A TERRIFIC WEDDING VIDEO!!! 
Something about having a dog (that’s not a service animal) in church rubs me the wrong way, but I’ll freely admit that’s my issue. If the pastor and couple are okay with it, why not? I agree that the potential for mishaps is great, but as long as the couple doesn’t mind that, more power to them.
We had our 2 retreivers as ring bearers. Worked out great. Granted, this was an outdoor wedding at a lake cabin.
Weddings are boring as hell, so even if the dog fucks up …actually especially if the dog fucks up …it will be very entertaining for the otherwise disinterested guests. I say do it.
p.s. I once saw one of those funny video shows where the dog in the wedding took a crap on his way down the aisle (he was the Ring Bearer). It was pretty funny and probably made for some great wedding memories.