24 hours later, a Support Ticket via Client Zone, another call, another WhatsApp conversation, an email with my ID, proof of address and a photo of myself with a handwritten note requesting a SIM swap, and there has been no movement whatsoever. No acknowledgement beyond an automated initial ticket response self-marking the issue as "Priority:
HIGH". I'd hate to think what "Priority: LOW" gets you.
To make things more fun, the 0800 Customer Service number is, in fact,
not toll-free. So, apart from all my partner's free minutes, I've now also spent about R50 in airtime re-hashing the issue for the 8th time, only to be put on hold.
At this stage, I feel like abandoning the number and starting a new life with a new number

You cannot make this up.