I was hopeful that would work, Adam, but it seems that it just wouldn't let me change anything during a postback. What I ended up doing was very hackish, but it works. When the dropdown list changed and triggered a postback in the InsertImage.aspx, I have it testing for the value in the dropdownlist. Depending on the value, I appended a new argument onto the querystring and then called a Response.Redirect to the new URL. In effect, each time the user selects a different item in the dropdown, the page is refreshed twice: once for the postback, then again for the Response.Redirect to get rid of the PostBack state.
Is this something done by design? As best I can tell, the CuteEditor.Dialogs.FileBrowserPage class is discarding any changes to the security settings if the page is in a PostBack operation. The coding is painful and it really slows down the operation, but it will have to do. The insert pages are just too complicated to try to reproduce in a custom dialog, so we'll have to settle for this.