I've found that by siding with the King I get the silver blade, and that by siding with the prince I get the gauntlets of crushing (or vise versa). How do I get both?

The silver blade? You get that from Saemon Havarian right before the ship capsizes. I side with the king. From siding with the prince you get whatever's in that little treasury closet. From siding with the king you get the prince's component of the 'Wave' halberd, as well as the gauntlets of crushing as a reward from the king, and the contents of the treasury.