You're skeptical that a far right Russian political network might have strong military connections who would have access?
I would also be skeptical of this idea if I truly believed that Putin had the extreme iron grip over military/FSB that he tries to portray. I believe it's quite a bit more chaotic there than he'd want you to believe though. I'm sure their opsec is not as good as it is in the US. After the SU fell they even auctioned off arms.
As to why not a gun - 1) it's quiet and surreptitious. 2) I reckon it might also be a message TO putin 3) I'm pretty sure they would want putin to be seen as responsible for the murder.