Sadly, their response suggested that the latter option is more likely than the former. They even had me fill out a form that asked how I'd like them to proceed--i.e., should they send another copy of the game or give me a refund? I chose "give me a refund," so hopefully they'll do just that in the next few days.
Will I ever see this title screen on my TV?
As soon as I receive said refund, I'll order another copy of Xenoblade Chronicles from a different retailer. I've heard good things about game.co.uk, which is selling the game (here) for just £29.99--£34.99, or about $55, if you include shipping to the States--right now, so I'll likely go with them when all is said and done. Here's hoping things go a little more smoothly the second time around...
See also: 'My copy of Xenoblade Chronicles has been despatched' and 'I just pre-ordered Xenoblade'