I would be sure to let the Attendant at the ticket counter & gate know your situation, they will probably have an answer or solution. SW employees are the BEST!!! I have seen many people trade seats if asked & usually no one wants to sit with someone else's kid. A long time ago, DS & I flew to need DH in MCO, we were on a flight that stopped in BWI & continued on. When we checked in ticket agent put us on the direct flight that hadn't left yet & called up to gate to tell them we were on our way. We boarded last, but were able to get 2 seats together through a swap arranged by FAs.
I don't agree that it is always the parents that cause e problem with splitting up. My DS12 will NOT sit next to anyone he doesn't know, regardless of the person's age, ie. He wouldn't sit next to an unknown 12yo. He loves to fly SW because in 25 trips, we've never had an issue finding seats together! Our August trip on JB, only had 2 seats together at purchase, 1 separated by 1 row & rest were extra legroom, which cost $35 extra, he was so freaked out that he wanted to stay home rather than go on vacation & I kept telling him no big deal, 1 row, etc.....no dice, he was totally stressed over idea. Luckily, at checking 24 hours prior, someone had changed their seat & he was able to sit with DH.
I would speak up over & over again, & ask about the boarding pass swap as well. Idk if ticket agent could, perhaps, change your number & DD, but you don't know what is possible until you ask & explain situation nicely. You will see TSA at gate, if they are going to check everyone, which I have personally never seen, but anything is possible. I would sit way towards the back as most people like to be upfront & you will have more people looking for seats there if you are trying to save.
Good luck! Have fun!