I think the difference is that Advanced Labs is a culture test, so they want you to be off of antibiotics so there are as many live organism around to culture as possible.
On the other hand, for the standard, antibody based tests (e.g. Western Blot) a course of antibiotics may help boost antibody levels (increasing chances of a positive test) by killing the bacteria and releasing more antigens from dead bacteria into the bloodstream to stimulate antibody production. I believe that is the theory, anyway.
I cannot think of how any supplement could increase your chances of a positive culture test. The best advice I can come up with is to try to time the blood draw for when you feel your sickest, as this is likely to be the time of highest presence of live bacteria in your bloodstream. If you have a fever, even better.
Hope that helps some.