Update: I got through again late tonight. After lots of wrestling with the booking, the agent (again in South Africa) put me on hold, purportedly because he needed to speak to Auckland, as he apparently couldn’t properly open/view the ticket himself.
Unfortunately, after a further 20 minutes, he checked in with me a couple of times and then apologised, saying he was actually on hold himself waiting to speak to someone in Auckland. So I was on hold with someone who was also on hold.
Ultimately he confirmed my mobile number and agreed to call me back within an hour, after he’d spoken with Auckland. I waited up - but he didn’t call back. 
