I'm with Wellbutrin here. The tune is most commonly done in C throughout. Without a detailed arrangement for your group, or a lengthy discussion prior to playing it down, it's probably the safest. Or if your leader can communicate key changes on the fly...
The Ellington/Strayhorn version probably went through several revisions or incarnations in its lifetime, so the original may not even be the definative for big band. While it can be fun to try and distill those arrangements for a small group, the focus of small group playing has always been more on improvising and less on arrangements, with the reverse true for big band. So it's best to keep it simple.
One alternative that I've encountered is to play the whole tune in C, then modulate to Eb for the shout, for the first half, blow over the bridge and melody out for the last A in Eb.
The possibilities are endless.