Our housemom had a cat. It mostly stayed in her quarters except occasionally, when she would bring it up and let us pet it. It was never an issue.
The only way this would be a problem that I see is if the housemom just lets the dog run around the house. Otherwise, I don't think it will matter. Best case scenario, dog becomes a sort of mascot, worst case scenario no one even knows it's there.
One of the Sig Eps on my campus was blind and had an assistance dog (I think he made it into the yearbook photo and quite possibly their composite). That dog was amazing. I am in awe of service dogs. The dog was super cute, and guys treated him so well.
If you were talking to someone and mentioned Sig Ep and someone wasn't sure which fraternity they were, most people said, "you know, the one with the dog."