Nobody's got it all right so far, but the good news for the OP is that they can use RAC at their resort to check their bags and get their BPs. SWA now is participating in RAC at all WDW resorts except for CBR and OKW.
If you use DME on the way in, RAC should check you in automatically sometime at or after the 24 hour mark. I personally would check myself in or use EBCI.
Take your bags to the RAC desk anytime after 6:00 a.m.--since the earliest you can use RAC is 12 hours before your flight--to check them and get your BPs. That way you can head straight to security once you get to the airport.
As far as getting yourselves to the airport, you will not be able to use DME as your pick up time would be at approximately 3:00 p.m.
That means you will need to get out of Epcot ASAP after the performance and probably just catch a cab. You'll have to drag your carry ons with you and stash them in a locker or something. One option might be to get to BC with your carry ons in the AM (probably via cab from ASMu) and leave your bags with bell services there. Then you could exit Epcot via the IG and walk to BC after the performance to collect your bags and take a cab from there.
FWIW, you really don't want to be getting that cab any later than 4:00 p.m. from wherever you are at the time.