HI Bob
We've often taken our golden retriever across the border (day trip) at Peace Arch crossing. Whether it has to do with our Nexus cardholder status, I don't know but we've always had the ownership papers and vaccination records with us, and of most importance to the border folk, proof of rabies shots. Even though we've never been asked, we've been sure to have those papers ready, and they will come with us this summer too.
As for RV parks and dogs, maybe I've been lucky, but on both sides of the border (along TransCanada and I-90), I've never encountered a campsite / RV park that was not dog friendly.
Sure, one has to keep dogs on-leash and no aggressive / intimidating types (rottweilers, dobermans, pit bulls etc) nomatter how well trained, but that's only fair to other campground users. Some do have dog walks or fully enclosed dog off-leash areas, so keep looking - they do exist.