You got it easy !!
When you have the trim off the drum, put a piece of 220 sandpaper around a small cleanning sponge. with a little pressure, sand with the grain in long strokes to remove the top coat that has faded. The paper will become fine the more you use it, wont take but a half hour, then before you re-install the trim, re-apply top coats letting them dry in between (12 hours). If you go with the same satin finnish, at least two coats.
If you use liquid cleaners, you run the chance of makeing the wood darker, uniformly darker, but still darker. Light wood gets darker when treating it and dark wood gets lighter. The guy who finnished it, did it the way I posted up top !!
You may get lucky and have particle accumulation and a good wood cleaner will do, but it looks like UV fadeing to me and in that case, remove the old finnish and apply new.
Remember, you`re changeing someone`s work. Are you gonna use tube lugs ?