There are many Fishing Bridge lovers here, but there are other options. For starters, Fishing bridge is just off the Grand Loop road in Yellowstone. Kind of by definition, there is some point on the Grand Loop that is as far away from Fishing Bridge as possible, so "centrally located" is a myth.
Another option is West Yellowstone, MT. There's a KOA there as well as
Grizzly RV Park.. Both offer FHUs, WIFI and cable, and at least at Grizzly, picnic tables laundry facilities and a playground for the kids. 12 miles from the west gate to the loop, but you're in the park for that 12 miles. One day, I spent the entire morning taking pics of a Bald Eagle family, a Fox and her kits, a herd of Bison crossing the road and a couple of Elk grazing along the road before I ever got to the Grand Loop. A plus is that West Yellowstone has retail grocery stores and gas stations where prices are cheaper. And West Yellowstone has the Wolf and Grizzly Discovery Center, where you can get as close to both as you really want to.
It just depends on where you want to be after a day in YNP.