I have spent nearly two years and have had hundred of conversations with my idols trying to identify a bully.  And have completely failed to do so.  And now - I seem to have a second bully.

This is utterly <censored> ridiculous and completely <censored> broken. 

I don't know if this is sure-fire, but here's what I do:

1. Find-out what clique the bullied person is in. (Socialize -> Talk about the group -> Relationships)
2. Get gossip from the other members of that clique; I go for the leader (Socialize -> Talk about the group -> Gossip)
3. Be annoyed about who's doing it.

Entire cliques will bully one person, hence there are multiple bullies. If you can get the leader of the clique to stop, the rest will too. You can also use influence to tell one of the members of the clique to let-up (which may in time eventually have them stop bullying, I think).

How you go about it is up to you. You can transfer influence (at a loss) from one idol to another by asking them to get blackmail on another, though that may fail and you'll be out the influence. You can move the leader of the clique to a different group (this is the most direct and quick method if you don't have influence over the leader), and I think you can also hold scandal points over their head too, though I've never tried.

Your #'s 1-2 are essentially what I was doing, almost every week, for two freaking ingame years.

Wasting the players time is not fun.
Forcing the player to pray to RNGesus is not fun.

Something in the mechanism is fundamentally broken.

You don't want to ask members of the bully's clique, because if anything they'll cover for one another.

The people to ask are the friends of bullied girl; her best friends preferably, if she has any.