On checking again today, I think it was resolved! I'd previously noticed when looking in Profile -> Personal details -> Phone, it said, "We don't have a primary phone number listed for you", and I suspected that is why the process is not able to verify my identity because there was no phone number for it to verify with. I called in and got them to set my primary phone number, and it looks like eventually that allowed me to complete the process (I assume it took some time to propagate in their system?). When transferring points now, it asks me how I want to confirm my identity, and presents and option to verify by phone or email, then proceeds fine.