I understand what you're trying to say, but in your specific example, it's more like 15/1000 would be sent on for confirmatory testing (western blot), of which only about 1 in 10,000 would be false positives. In your scenario, this screening of those who request the test but otherwise do not need it would result in statistically very few false positives, on the order of 1 in 175,000 of those who make the request.

Perhaps a more likely error would be among those who request testing for Lyme disease (currently a popular self diagnosis) and thus get put on an unnecessary regimen of doxycycline. I'm racking my brain trying to think of an example that would be much, much worse. Perhaps testing for antinuclear antibodies and getting diagnosed with lupus?