My advice would be to rent this outside of Chicago. The big tourist bike rental operations charge $70 a day for a tandem. I found a few places in Iowa charging $110 per week. That said, I am sure what availability will look like for the dates you need... the RAGBRAI forums might be more useful.
Also, when renting a bike, you can always bring your own pedals if you are concerned the shop won't be able to meet your needs.
Try contacting the Quad Cities Convention & Visitors Bureau at (309) 277-0937 or visit their website at www.visitquadcities.com. The rent tandems and promote cycling in Davenport (which is near the end town of Clinton, IA).
You might consider buying a 10 year old Trek or Cannondale and reselling it afterward. The advantage being that you get a chance to set it up to suit and get some experience on it before the ride starts. If bought for market value, you should be able to resell it for no loss. If you do some cleaning and tuning you might even make a profit.
How about a tow bar? Tandem when you want to ride together, 2 solo bikes when going your own way?