Okay, I've got it working.
One of the attributes of the CollapsiblePanelExtender is SuppressPostBack, which you can set to true or false. Before adding the Cute Uploader I had an UpdatePanel within the target of the collapse panel extender, and I had SupressPostBack set to true. So expanding and collapsing the panel was nice and smooth. When I added the Uploader, I removed the UpdatePanel, but I didn't get rid of the SupressPostBack=true. So there was no postback happening, and the Uploader could not fully function.
So, in short, the Cute Uploader works fine within an AJAX CollapsePanelExtender. Just make sure a postback can occur.
Thanks for your quick response, and simplified, working example. Very helpful.
Dan