So... I found someone at Vodacom willing to help me sort everything out. The root of the problem comes down to mistakes made at the company (understandable) and the fact that most staff have severe tunnel vision and can only focus on one thing. For the most part, nobody "owns" a problem but focuses on their specific processes. If they assigned each person to an account manager responsible for overall customer satisfaction, such problems could be avoided.
The department that eventually addressed my problem was "New Sales", but only because somebody was willing to step in and liaise with all the other departments to resolve it - most people in the same department just said they couldn't do anything.
Unfortunately, we sometimes have to deal with companies we don't necessarily like. There are only two significant mobile operators in South Africa - Vodacom and MTN. The others are wannabes who don't have the resources. I have dealt with MTN for many years, and they are no better. While some might question my desire to subscribe to products at a company that offers this kind of service, I do not have a choice. There isn't any alternative. It's like telling someone to avoid Eskom because they are a horrible company - we all know that.
I switched to Vodacom because MTN does not offer roaming in some places and charges exorbitant rates in others. For example, a call on MTN from Serbia costs R150 per minute, while on Vodacom, it costs R5. MTN offers no service in many countries, primarily those that were part of the former USSR. This is possibly due to the political issues stemming from the country's close friendship with Russia, but Vodacom seems not to care, so I'll stand by them on that point, at least.
Naturally, if the problems were with my credit, any refusal on their part would be justified. I only have utilities, insurance and credit cards on my profile. All of these are paid promptly every month, and I pay my cards in full, unlike irresponsible people who carry a balance at a tremendous cost to themselves. I knew I was creditworthy. Otherwise, I wouldn't bother fighting it.
Regardless, thanks for the advice and suggestions offered. The usual customer service issue is finding someone willing to do their job, and I found someone, so the matter has been resolved.