@Helgi Kremlin love "multi-faceted operations" or "mnogokhodovochka".
So, it can be made for multiple goals at same time, including "represent Russia as victim of international terrorism" while Kremlin is the main control center of international terrorism, e. g. hamas, ISIS and alike, and all that other goals mentioned earlier.
Your question on why they used that group but not other one, reminds me FSB complaints I seen in media that "it is hard to work within ethnic diasporas" for them.