Back when I was a finish painter for for a contractor who specialized in high profile custom home interiors, I often had to chance down scuffs on trim and cabinets left by other other types of construction workers. They’d bump in to our fine enamel work and often leave a mark of dirt or some other form of color transfer. If the stuff wasn’t accompanied by a paint chip or a dent, I could often repair the area with a small bit of Soft Scrub in a soft rag. Going over the scuff using light pressure in a a circular motion, followed by wiping with a clean wet rag would restore the “damaged “area.”
Its possible Soft Scrub might work in your situation. If you have some handy, you might try it on an inconspicuous section of your frame.