I don't understand why this woman is making her mother choose between her and her baby or the sister and her dog. A 6 month old baby hardly needs much supervision and will be asleep for most of the time anyway. Surely the baby's parents don't spend every hour of every day watching it. So baby could be in bedroom asleep (with monitor) and grandma can spend time with dog. I think obstacles are being thrown along the way just for the sake of it.
It will be a different matter when the child starts crawling and toddling, but at 6 months old babies are rather boring and don't do much other than sleep, eat, cry, piddle and poo.
I am a great dog lover, but no dog - no matter how well trained, or not - should be left alone with babies or young children. So regardless of who is doing the supervising they should always be separated when there is no adult around. Children and animals in general are not a good mix. Babies smell very different from what they are used to, they make weird noises and dogs in particular are curious about them. Let them have a sniff and get used to this new thing while adults are there. The worst thing you can do is snatch baby away as soon as the dog appears. As soon as the novelty wears off the dog will generally go off and do its own thing. If baby is going to sister's house do not relegate dog to kitchen/garden etc. It is the dog's home and the baby is the interloper so why should a member of the household be banished because of a visitor, it is a sure fire way of causing a rift between them.
Is the dog really as badly behaved as is made out? or does this woman just not like dogs?
Why would you choose to get married when you have a 6 month old baby to look after?
What is the baby doing on the wedding day?
The whole thing sounds as if these people are both reckless and feckless and don't put a lot of thought into what they are doing. If it were me I would have either postponed the wedding for a few years (see if the relationship lasts that long) or cancel it altogether. If the father/prospective husband is so selfish to go away for a weeks holiday with his mates this close to the wedding day, will he change once he is married?
P.S. I hope you got to your interview and that everything went okay.