If the wood will be painted it might work with a ‘hole saw’. Saw a rond hole removing the anchor and then saw the same size hole in a different piece of wood giving you a wooden disc. Glue the disc in the hole and sand it is completely until you can’t feel it anymore. Paint it so it is also not visible.
Do be aware that this is very hard to do and it might not look as nice. Try it on some scrap first.