It's a safe bet that Prince Corporation from Holland Michigan built your visors due to the fact that back then they held the patent for illuminated visor vanity's beginning in the late 70's. With this patent, Edgar D Prince became an extremely wealthy man and built the Prince Empire that was later purchased by Johnson Controls in 1996. I began working there in 1984 and at that time scored a few sets of illuminated visors for G bodies that I owned. My 442 still has a set and is likely the only LH and RH illuminated vanity with the correct substrates in existence for a G body. An illuminated drivers side visor was never an option from Oldsmobile.
To answer your question, you can stitch through the polypropylene substrate with an industrial sewing machine but the visor core substrate needs to be opened up and re sealed with the fabric pinched between each half before sewing. The backbone of the visor where the elbow is inserted has a living hinge that will likely crack now after 30 years but its really not an issue as you can glue it back together. Before you attempt to split the substrate, make sure to remove the lighted vanity pack first by removing the lenses and then the screws behind them. The rectangle fabric inlay in the vanity cover you're referring to is made from the same fabric used to cover the visor. It has the same foam backing on it but was sandwiched between the bezel and the cover and then sonic welded with a Dukane sonic press. Getting those 2 pieces apart without wrecking them may be a challenge. Maybe someone here has been successful.
