Aloysius would look very thoughtful at this point and say that there is another way they can undermine the Death Eaters and create dissent - rumours. The beauty of rumours is they don't have to be true, they just have to sound true.
The rumours could be almost anything - from rumours about the activities of the pureblood women to rumours of wrong-doing of the pureblood men to rumours of corruption in the Ministry. Not all of them will be believed and some might be able to be debunked but there's a damn good chance that enough will be believed, not just by the general populace but also the Death Eaters themselves, to start some suspicion and infighting amongst not only the Death Eaters but also the purists.