I do not have a Southwest card, but I can change the closing date for all of the Chase cards that I do have. You don't change the closing date directly, instead you change the payment due date, which is 25 days later for personal cards. I have all my personal Chase cards set to close on the 3rd, due on 28th. To close on the 15th you would probably choose 10th as a due date, but wait for someone to confirm before to make the change, if the exact date is important.
To make the change:
Login to chase.com
Choose the card account you want to change
Click: Things you can do > Update Settings & Preferences > Payment Due Date.
The change is not effective until the next billing cycle.
Dang. I was hoping I could get them close it early as I've already done the required spend and the sooner it closes the sooner I get my CP.
Thanks