After approx 10 phone calls to both Lufthansa, and Avianca Lifemiles, eventually I got it rescheduled.
Lufthansa had to make a new booking under the same record locator for the new flight (same flight number, origin, destination ... different day). Lufthansa then told me to call Avianca to "reissue" the ticket (but at this point, the new information was already loaded into the PNR for them).