You’re correct that Fishing Bridge is the only cg with hookups and also that some cg’s allow generators and some don’t. I’ve been to Yellowstone many times with a PU and a 16’ TT and only twice have I not gotten a site. Once I boondocked outside the park and got a site in the morning. The other time I got a spot in Fishing Bridge. That was a bummer, I felt like I was in an RV storage lot. And I found another site early the next morning. And almost all my trips were in peak season, so I don’t think you’ll have any trouble at all getting a spot with a TC in September.
I dry camp, too, and what I usually do is take my battery out and have it charged at a local station. I’m gone all day sightseeing anyway so that’s never been a problem. In another park I might move to a cg with hookups, but I won’t do that in Yellowstone.
If you’re in Yellowstone I’d also go to Grand Teton NP. My description is Yellowstone is beautiful, the Tetons are spectacular. The glitch is that a week isn’t really enough for both, you can easily spend that much time in Yellowstone. It’s huge and I put 400 miles on inside the park on my last trip. If you go to the Tetons check out Lizard Creek CG. Small sites, some on the lake, and they have a no generator loop. Enjoy your trip.