I would think paint thinner / paint remover would do the job? Also, research Oxy cleaners? Try Go Gone. Even try Magic Eraser.
I would bet if you spend enough time, you could find a thread in here about this very thing - cleaning paint off of satin wrap.
Be sure not to get the wood wet though (perhaps preaching to the choir?)
Above all - be sure to test these before using widely - to make sure they do not damage the wrap you are trying to save preserve.