From a legal and regulatory standpoint, if you are board certified in AP, CP, or both you qualify to be a lab director.
Would I hire a fellow in this situation--never. And as the point is made above, I'd be extremely wary of someone who would hire me were I in that position.
I know of a couple fellows who have done it if board certified in CP. They have been medical directors of several national plasmapheresis centers in our area. The work basically entailed doing donor chart reviews on the weekend and making sure that the center meets all regulatory requirements. I wouldn't take on anything more complicated than that if you are a fellow.