Took another look on RedWeek.com again, and the place Rob's staying at is available. It's $3,400 to rent for the week I need. That's a whole lot more than the under $150 it would be for a trade, but if a trade doesn't come in, I don't know what my options will be.
I just added the Lawai Beach Resort to my Interval search, as well as the Marriott Kauai Beach Club, although I don't think I'll be getting in any Marriotts through Interval; it's really hard to trade into a Marriott when you're not a Marriott owner.
There are other timeshares on the island, but I'd like to be as close to Rob as possible. Not sure if it'd be better for my family to just go anywhere on the island, or or just pay the $3,400. Who knows? Maybe I can find a place on eBay.