Region-free 3D blu-ray on Amazon UK: "This item does not ship to the United States"
Problem solved!
If anyone else gets this problem... Go to the additional listings, and find the one sold by Amazon.co.uk, instead of the default seller.
I picked that one, it even said it cost 18 pounds, but when I got to the checkout, it suddenly only cost 15 pounds, which should have made it the default option.
Cranky site! But this one will ship to the U.S.
YAY!