For those interested in the followup on this...
I decided to take a simpler approach. The "wall I ran into" was not simply that the CuteEditor was appearing within a nested user control, but that the user controls were being dynamically added to the ASPX page. They therefore simply do not exist on PostBack, so there is no control (i.e., CuteEditor) to which I can get an object reference. Put another way, the FindObject method has no container within which to search for the CuteEditor.
Note that this has absolutely nothing to do with the CuteEditor.
If I was not dynamically adding the user controls to the aspx page, then I'd be in business. So, if you are declaratively including user controls on an aspx page, then you could use FindControl() to get a reference to the CuteEditor that exists within the user control.