I would try to press it outward slightly from the inside, with some rounded blunt object if you can't get your hands into it, and then I'd put a small fillet of wood glue into whatever crevice remains inside with a small patch of the glue glaze over the area. You may have to get creative to apply it. I might also put a surgical fillet of wood glue onto the crack on the outside. Use a toothpick or an old pencil point or very small artist brush you don't expect to use again, and maybe thin the glue. Do one side at a time and let dry. There will be a small mark but I don't think the crack makes a lot of difference in the playability.
Some clearcoat or varnish might work as well or better than the glue, and you may be able to "spot" match the color of the wood to minimize the mark.
Clearcoat spray paint can be sprayed into the cap and brushed or spotted on with a point. Don't blotch it all over the outside though.