I think the100thMonkey is onto something, at least for small systems. Murder the other factions ships, especially the controlling faction. At first the anarchy faction won't gain very much influence, but as its influence grows this will speed up.*
Once you get the anarchy faction above 7%, either push it up, or another faction down so they are equal. Provided neither faction is in a conflict somewhere else, they will eventually go to war. Anarchy factions never have elections, it's always war or civil war. Once the war starts, gaining reputation is easy by fighting for the anarchy faction and cashing in their combat bonds. It will take about 7.5 million in bonds to become allied.
Expect to take at least 2 weeks, and to be wanted for this whole time.
* what i'm referring to is: If you have 4 factions, A 50% (controlling faction), B 40%, C 9%, D 1%. If you knock 10% off faction A, the other factions will gain: B+8%, C+1.8%, D+0.2%. So not so great for faction D.