Kennel or crate, I know you said you don't have/want a crate but we have not talked about a kennel. " tried all the usual stuff" we never know what all that entails so be patient with us, we may repeat something your already tried. Dogs usually love snow, it's novel and they don't shovel it like us so its fun. Mud and wet surfaces can cause them to be cautious and less likey to hold it until they go outside. Outside may be unpleasant for them but I suspect its something other than mud. Nothing else new, right?