Very few WDW TS locations are equip to handle take outs in the traditional sense. In most cases, you cannot order a meal the podium. You have to be seated, order, and then ask for it to be boxed up. So, you can try asking at Beaches and Cream if they can do a to go order, but be prepared to be told that they cannot accommodate the request at that point. I would keep trying for a lunch reservation; people's plans do change, so ADRs do come open.
Beaches and Cream will not do a to-go order. They do have a takeaway "lane" that offers beverages and ice cream selections - no food. Hurricane Hanna's at the pool will do made to order burgers if that's what you are after.
Like a PP mentioned, Hurricane Hannahs has burgers, and I've heard they're exactly the same as beaches and cream. So you can order food from Hurricane Hannahs and sit there (or at the beach, or at the pool if you're staying at YC/BC) and then get a sundae as a take out from Beaches and Cream. Should get you basically the same experience that you would have gotten if you'd done take-out from beaches and cream.