Suggestions for 2-3 Week Road Trip (LA to Philadelphia)

Hello,

My wife and I are going to move from Los Angeles to Philadelphia this July and would like to spend 2-3 weeks on the road visiting national parks along the way. We were thinking about driving through California, Nevada, Utah, Wyoming, Montana, North Dakota, Minnesota, Wisconsin, Michigan (I have family here), Ohio, and Pennsylvania. This will give us a chance to see Zion, Yellowstone, Glacier National Park, etc. We will have a ~1 year old goldendoodle dog with us and unfortunately the majority of national parks are not dog friendly. Would you recommend visiting all of the national parks along the way even with a dog, or are there alternate places to visit that are more dog friendly?

We're also torn between taking the northern route (Yellowstone -> Glacier -> Theodore Roosevelt -> Voyageurs -> Isle Royale -> Upper Peninsula) and taking the southern route (Yellowstone -> Mt. Rushmore -> Badlands -> Voyageurs (?) -> Isle Royale -> Upper Peninsula). Which route would you recommend? Additionally, do you have any recommendations for improving our trip and are there any sights / national parks that we must see? Our trip is extremely flexible so all feedback is appreciated!

Thanks!