Thanks, everyone, for your tips and inspiration. I did my basic adjustments in LR, went to PS, selected the shuttle with the quick selection tool (then significant edge refining), processed the shuttle primarily for brightness, contrast, and color. Then I inverted the selection, increased the exposure of the non-shuttle area, then went back to LR and used a brush to slightly lighten the people. It's the most significant processing I've done for a photo. I might still do a little more tweaking, but I couldn't have gotten to this point without seeing the results you folks were able to generate. Thanks again!
FYI, the Endeavour is at the California Science Center in Los Angeles. The room you see here is temporary. They are breaking ground on a new building that will house the shuttle in a vertical orientation. The orange liquid rocket and the two solid fuel rocket boosters will be attached just as they would be for takeoff. They will have various balconies all the way to the top that should provide incredible views and photo ops. It will probably be a couple more years before that's completed.