I was bullied as a child pretty relentlessly from the age of 13 onwards. Made me suicidal, so I know bullying from the receiving side of the equation. Hence my desire not to excuse bullies in any way, which your initial post certainly seemed to do. If that wasn't the intent, then perhaps that's my misunderstanding. It was perhaps the statement that most bullies are just reacting to being bullied at home. This isn't always the case...but then perhaps our experiences in this regard have been very different.
In terms of expressions of primal rage against children, I never expressed any rage at any child. I do, however, think that the parents of the child in the OP need to be made aware of and responsible for the situation. That way, perhaps there's some chance of resolution.
I had assumed, as had most of the other people on this post, I'm sure, that the the OP was kidding when she said about her DP going round and threatening the bully's dad.
And it's an urge I certainly understand having endured 3 years of pushing and shoving that finally ended when I got pushed down the stairs by several of my class mates.
These children were not from bad homes. As Frodosgirl points out...not all bullies are. They were just arrogant, stuck up and got caught up in thuggish, gang-type behaviour.
I was the one from a rough background, laced with physical abuse, but I, strangely enough, felt no need to bully others. And these children were all 'well to do'...from the nice/privately owned houses. Hence their 'unexpressable rage' at my doing well and taking the valuable top class place of their equally well to do, but unfortunately somewhat dim, friends!
The results, regardless of any intent, was almost fatal. Regardless of this, I have no rage, but just rather well grounded dislike for bullies.
I hope, by the way, that you managed to sort out the problem with your own child. It's the most unpleasant situation to be in and I can truly empathise...