The approved hospital thing makes sense (it is in the legislation), but I find it impossibly hard to believe that there isn't such a facility in SA.
I was told by everyone when first starting that I would have no choice but to go through a gender clinic, because they were the only people who would deal with trans patients, but that just simply isn't true. I would go with your gut feeling on this one; it certainly seems like they're trying to steer you to me.
Besides all of that, the surgery that you want isn't even strictly 'trans-related'. It's not like top surgery, or any kind of reconstruction (unless you wanted it all done at once?) so it really shouldn't be an issue in the first place.