While the rain might ruin some traditional camping, going a bit more upscale can certainly be just as exciting. Even if dogs aren't allowed on the Theodore Roosevelt National Park, that doesn't mean you can't enjoy the beauty around it from the comfort of a rental or Airbnb. Stay inside while it rains, make your plans for when the skies clear, then launch out into it without having to go anywhere.
North Dakota may not have many food claims throughout their history, but there are two perfectly good dishes to share between the two of you: Bison Steak and Pitchfork Steak, both of which the state is known for. Although high levels of salt aren't great for your dog, a few tasty cuts off a well-cooked steak will likely do nothing more than give them a warm feeling in their heart and belly (and yours too).