So after trying to phone about 30+ times this week, I finally got through (after waiting for 56 minutes) to an agent who said they've managed to enter the secondary contact info on my behalf. At that point the 1 hour was up and the line disconnected.
I couldn't have cut it any finer.

Still no idea what the cause of this issue was.
It is frustrating that all it took was about 3 minutes on the phone with an agent to get it sorted, but it is so difficult to get through to BA at present.
Live chat isn't going through and they are taking days to respond to social media queries, often with irrelevant replies.

Next 1st world problem: figuring out how to prebook a standard vegetarian meal as a non-status passenger who will otherwise likely get served last in a full club world cabin.