Are we forgetting something here?
Who owns the house? If OP does not want the dog, then he should enforce the rules NO DOGS! Or a problem dog that does not need to be in the house. I would not go against the issues of boundary and make chaos in the household because everyone feels sorry for the dog.
If OP wants to keep the sis' dog out of kindness, then he should make way and make the rules for everyone to follow so they all will be on the same page working with the dog's issues. If no one is doing it, then OP has the right to see what's best for his home, his family and his pets either to kennel him outside with a doghouse and run OR give him away/shelter. So if you want to keep your house looking new, better put your foot down. It is tough love for everyone. Like the mud room, if that is the place that he is allow to make his mistakes while training, do not give any other rooms in the house for any opportunity for him to mark his territory.
Set boundaries for everyone, yeah, they would get mad at you for having a hard heart but you didnt have inside dogs for awhile until now. If you want to keep the dog outside, wait until spring and by next winter he will be adjusted to outside living. Be mindful about getting a heated pad for the dog because I do not think the wired haired dogs have enough sufficent down coat to keep them warm until you can get a dog coat for him.