How the heck do I get the metal buttplate off!?  It's somehow connected with the sling swivel, but I'm not sure how.

You need to press down against the buttplate to compress the springs.  This will remove tension from the sling ring.  You could try a strap clamp or a C-clamp with a couple scraps of wood to keep from damaging the stock.

Use a pair of pliers to gently pull one side of the sling ring out of the post.  When you get one side out, let it bend back into shape but keep the end out towards the side of the post.  Then just slide the other end of the ring out of the post.

Excellent, worked like a charm.  Thanks, I owe you a beer!