I will suggest that you see if you can get at least a couple of nights in Fishing Bridge reserved, and also plan on a couple of nights at other campgrounds. If you can go a day or two without electric, you should have no problem getting into one of the other non reservable campgrounds
within Yellowstone. In other words, don't commit to one single campground for the entire time, and do yourself a favor and stay
at least a few nights inside the park so that you can go to sleep and wake up
in the park. There is no better time to see the most charismatic wildlife than in the early morning or at dusk.
Yellowstone is HUGE, and by moving your base camp from region to region, you can make the most of your time. If you stay outside the park, you can experience very long wait times to get in, especially mid to late morning. It can also take an hour just to go a few miles due to "animal jams" or construction, and even if you aren't stopped for some reason out of your control, the speed limits are very strictly enforced, for your safety and that of the wildlife, and you will want to stop to see the sights, and you just don't want to waste your time. Distances are vast,
see map, for example, it is about 14 miles from the west entrance (by Grizzly RV park) to Madison, another 14 miles from Madison to Norris and another 21 from Norris to Mammoth, or in the other direction, 16 miles from Madison to Old Faithful. So, from the west entrance to Old Faithful, it is 30 miles one way. That 30 miles can take 3 hours to travel due to a few animal jams... Plus, you might get stuck in line at the entrance for an hour or more. Now, imagine instead, that you have camped at Madison...
I highly suggest not using a single place as your "homebase".