I tend to use the term “paramilitary wing of the Democrat party”, but yours works, too.
You are quite right, and the term "paramilitary wing" is more succinct also.
I think many younger people don't realize what the "paramilitary wing" of a political party does, since the dirty war in Argentina and the various Communist paras of Central America have disappeared, or at least receded from our daily consciousness since the fall of the USSR. That's why I used the terminology I did.