Have you ever tried allowing W to have the dog even for one day? You really don't know until it has been tried.
Has W had the kids over too? I mean...do you see it at all?
Your so-called logic has me baffled. Just doesn't make any sense to me at all. Your W knows that the dog has medical issues. Who's to say that her place isn't any better than yours? Dogs really don't care where they sleep as long as it is a warm place.