At the time that I am writing this, SkyAuction.com has some Ready-Set-Go availability for Sept. 2 (1 week) for a 1BR condo (sleeps up to 4) at either Silver Lake Resort (just behind Animal Kingdom in Kissimmee) or Cypress Pointe Grandvillas (about a mile to Downtown Disney in Lake Buena Vista). Both are Gold Grown timeshare resorts - very nice. I can readily recommend these resorts, having stayed at both in the last couple of years.
For the low price that you are likely to get (just bid next to nothing, plus the $200 add-on fee they always charge), you wouldn't have to worry about checking out a day or two early. You'd only have to pay for another hotel room on your arrival day.
If you need a larger unit (more than 4 in your party or you don't want them sleeping on a living room sleeper sofa) or if you need a place for that arrival day, HotelKingdom.com has Cypress Pointe 2BR condos available for $93 per night. They list Cypress Pointe Grandvillas (same resort, different section) 1BR condos for $71 per night. For 6 nights, this is just $426 - a pretty good deal for what you'll get and a great location.
I agree with the Priceline recommendation so long as you only have up to 2 people traveling - they don't guarantee a larger room. Visit BiddingForTravel.com to learn how to bid properly before you try Priceline, selecting a higher star rating and only the area(s) that would be acceptable to you. Remember that all of the bidding sites (ex., Priceline and SkyAuction) are non-refundable. HTH.