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 making a 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.
The statement ending date and the payment due date are usually 3 days apart. Yours is different because one is at the beginning and one is at the end of the month. In Chase's eyes, day's 29, 30 and 31 don't exist since these days do not exist in every month. This is because they are required to keep your payment due date the same every month. As far as the 25 days is concerned, that is the minimum. It will very depending on how many days are in the month.
The OP will need to change her payment due date to the 12th if she wants the statement to end on the 15th. The change may take one billing cycle to complete and usually will extend your current cycle a little longer, not shorten it like the OP had hoped.