Am I crazy for wanting to buy $20 worth of Dove to get this?

Not crazy at all. I'm a sucker for freebies too. I bought a pair of shoes at DSW a few weeks ago just because I liked the free bag they were giving away with purchases. The shoes are okay but I love the bag.
 
My makeup bag gets replaced every couple of months because I can't resist getting whatever Clinique gift is being given out if it's cute! At least I always have travel size products for vacations!
 
You're not crazy at all! At least you'd be buying something you'd use everyday (lotion, soap etc) and the bag looks pretty functional too. It'd be different if you were talking about buying something you'd never want/use/need OR if it were more than $20. Go for it!
 
I don't think you can have enough bags that come in the price range of 'free' so I would go for it. If nothing else, you could keep your Dove in it. ;)