Assuming that the excitation source is going to be a ring of Am 241sources, the fluorescence x-rays will be a fairly diffuse region. As such, it would seem that an integrating sphere will less beneficial. I would think that you would want the target material to be in close proximity to the scintillation crystal. I would also think that you would want the scintillation crystal to be in close contact with the detector. The surfaces of the scintillator could be coated with a reflective coating with the exception of the window to the detector to maximize capture by the detector. I would suggest aluminum foil for the window to the sample. The side geometry needn't be thin and there would probably be an advantage to using a high Z material to prevent fluorescence from the target x-rays interacting with it. Gold foil as used for gilding purposes may be a good choice. Presumably the sides of the crystal will be further shielded with lead to prevent source x rays from entering the crystal.
I would think that the scintillation crystal would be thicker rather than thinner to permit greater interaction of the fluorescence x-rays.
I would think that the scintillation crystal would be thicker rather than thinner to permit greater interaction of the fluorescence x-rays.