Hi Brent,
Unfortunately your problem seems to be the one reported in CONTRIB-2135 for which there is no solution.
Mike wrote "Unfortunately, there is no way to do that. If you restore a course with a questionnaire that is "using" a public questionnaire, there is no way to verify that the questionnaire is is "using" is the same one as it was when it was backed up. Therefore, no connection can be restored. Instead, it creates a new questionnaire in the course."
Joseph