I ended up not using the Editor Widget. Instead I implemented custom editing using the TemplatePicker and AttributeInspector, and placed the AttributeInspector in a separate dialog. It would be helpful if ESRI would modify the EditorWidget to support this scenario, or at least document this issue.
BTW, I also noticed that the templates in the TemplatePicker do not draw if the div that contains them is set to display:none. So either I have to delay initializing the template picker until the user clicks to display it (which was suggested in another post), or use visibility:hidden instead of display:none.