I ordered the new flapper assembly, disassembled the ice dispenser assembly and installed the new part.
There are several good sources of assembly/disassembly instructions out there but one word of caution: The job is pretty straight forward but be very, very careful not to force anything. Watch where things are clipped. Most of this is flexible plastic clips and its easy to break them. I'm giving this job a 3 for that reason. You have to be patient and not force things too hard.
I broke one minor clip that did not effect things going back together because a screw does all the real retention work but I wish I was more cautious.