I've made the trips many times. It depends on what you might want to see along the way. Since the Northwest is our Jam, when we go to see Yellowstone, we've kind of seen most of the stuff along the way so we tend to drive to arrive, not smell the flowers along the way.
You are right to want to book your stay at Fishing Bridge in Yellowstone. The park is massive and distances between either end of the park, and between attractions is massive. Staying at FB will greatly reduce your drive time compared to having to drive into the park every day from somewhere else. As you've experienced, it books up well in advance and spots in there are hard to come by. That said, West Yellowstone is a charming little town with places to stay. But, you'll be in line with everyone trying to enter the park every morning and driving in traffic to the attractions.
You obviously need transportation inside of Yellowstone. If your camper/van is your primary means of transport, you will need to pack up every morning so you can sightsee and then re-deploy every night when you return. Remember to add extra wiggle room if you're planning full-day agendas inside the park! As well as traffic. Bison (and the people who stop in the middle of the road to gawk) really can cause jams!!
Don't forget the evening ranger talks and tours that the park does. The ranger staff are magnificent and we've cherished our time interacting with them while on our stay.
If you're stopping between Seattle and Yellowstone, we like the Liberty Lake RV park
The Best Liberty Lake RV Campground just outside of Spokane.
In Montana, we like Jim and Mary's RV Park in Missoula
www.jimandmarys.com
If stopping relatively close to Seattle (~2 hrs away) we like staying at Wanapum State Park
parks.wa.gov
If you are planning to also see Glacier, we have really enjoyed the Glacier KOA
West Glacier KOA Resort is located in West Glacier, Montana and offers great camping sites! Click here to find out more information or to book a reservation.
koa.com
If you're looking for a destination for the kids along the way, Silverwood in Idaho. Although I have not been in decades, it still looks wildly popular in the summer months (they also have RV sites there).
Discover the ultimate family-friendly destination at Silverwood Theme Park! With over 70 rides, attractions and shows, there's something for everyone. From thrilling roller coasters to water park fun, our park offers a day of excitement and adventure for all ages.
www.silverwoodthemepark.com
Those are just a few ideas...good luck with your planning!