To answer a few questions, no noise and no visible signs of what is causing the issue. When watching from the outside, the sticking occurs before the door is open enough to see if weather stripping is stuck or some other place is sticking. Basically, once the stick/suction breaks, the door opens fine, but it is sealed shut until whatever it is gives up.
I thoroughly cleaned every point of contact I could find and then hit the weather stripping with some 303, and the problem is about 90% better. It still doesn't open 100% freely compared to the other doors, but I no longer have to literally put my shoulder into it to get it to open.