Some of you guys are making this way too complicated....

a quick google search and I found the following websites which sell small quantities of seed cotton

Whatcom seed

Southern Exposure Seed Exchange

To the OP, notice two points:

a) notice in the Southern Exposure website, it says the cotton is 120-135 days.  Think about the season length corn you grow in your area.  That is going to be the problem.  As I mentioned above, if you start the seeds indoors, and then transplant them, you should be able to get around this.  I am presuming that cotton plants can be transplanted - assuming, as I have never done it.

b) the other point to make is that they cannot ship to Georgia, South Carolina or California.  That would be, I presume, due to the risk of boll weevil... as one other poster has mentioned.  However, since you don't live in S. Carolina or Georgia or California.... or any of the other states that commercially grow cotton, I can't see this being an issue to be concerned with.

To the O.P., if you and your wife are interested in trying it, I'd say Go For It.

Finally, on the Park Seeds website, I think I found the basket and cotton plants that you referred to.... or something similar.  Park Seeds sells a basket for around $10. Look at what they sell the basket with the cotton in it.