If there is movement, then it likely would have an angular component, a twisting moment may occur as it goes from the vertical to horizontal. Seems there may be quite a bit to measure. There are simple electronic levels in hardware shops that would give you your 'C' angle. They would not have the resolution to determine if the heads themselves were twisting on the end of the arm. It could also be that only one end might shift. In that case you could put a platform in the middle on the axis about which the 'C' arm turns (It would need to be rigid enough not to be affected by gravity itself) projecting out as far as the heads to support a target grid on either side of it, and place a camera on each of the heads pointing at the targets.
When you say you need to measure background radiation, is that just an aside, and you are equipped to do so, or are you needing advice on that as well. Not my area if so, but this makes and interesting read.
http://www-naweb.iaea.org/nahu/DMRP/documents/Chapter4.pdfThere are usually companies leasing out test gear for such purposes, the equipment should have a current test certificates issued by a legitimate testing source.
Cheers,
Richard