I own ZERO slabbed coins, so someone who is more familiar can maybe correct me if this is "run awaaaaaaay!!

" advice. But here is what I would do:
Write to NGC. Tell them you have a coin in one of their slabs that has developed verdigris and that you would like to conserve the coin, and inquire if they will waive the slab fee to reslab after the verdigris is removed. The worst they can do is say no.