I'm pretty sure I've never seen any transcriptions of any kind of music with fingerings included, meaning reasonably accurate transcriptions taken directly from the recordings. I doubt any exist.
Maybe in some primer or type of graded book there are some arrangements with fingerings.
If I'm understanding your example correctly, you're playing a perfect fifth (A + E) in the RH, then (C+F) quickly above it, then back again?
What's wrong with the way you're doing it now? If that works for you, I don't see why not. If I had some reason to do that, I'd probably use 1 and 3 for the A5 "power chord," and then maybe 2 and 5 for the bVI chord (or whatever it's supposed to be, with the C and F).
You can probably find some concert video of Carole King playing her tunes and see what fingerings she uses, if you want to go to the effort.