Not sure what you need a specialist for specifically, but for ~$300 you can get an appointment here in South Africa for any specialist you want. If I want to see a specialist, any specialist, I call their office, book, and X amount of time later, they see me. The reason I mention it is that SA has a medical tourist visa you can get, so flying here for your medical needs may well be within the realm of cost-effective for a lot of people in the Western world.

As a general aside. That you have to plead, motivate or beg your government to let you get healthcare from a specialist seems alien to me. I just don't even have the words - your government does not own you!