I give a light wipe with a damp towel (after a shower is good) after wearing, then air dry. Nude, unfinished leather I might clean with a very light dose of saddle soap (learned from my dad), wipe clean, air dry. All the leather straps I have is “finished” leather like many car interiors, so cleaners could damage the finish IMHO. Just the damp towel, pat dry, air dry works for my Hirsch and other straps.
Maybe a photo of yours would help. How “worn” is it?