I find it hard to believe that "pretty much everyone hates Bellatrix" since a ruler so disliked by its nation neither amasses followers as a revered prophet nor stays in power for long, especially when there's suspicion that she arose to her position via matricide/regicide, clearly a substantial group of people support her and her rule. At the same time, though, a lot of people are clearly unhappy with the current state of the Kingdom. Thing is, the only given faction that opposes the Queen right now is a sect of murderers with its own religious beliefs that seeks to destroy the entirety of the ages old government system. Interesting from a roleplaying perspective, certainly, but it's rather extreme, don't you think?

In situations like these it's quite likely that a group who opposes Bellatrix's rule specifically but doesn't want to do away with the Kingdom's monarchy in its entirety would arise from within the Kingdom itself. Groups like that are usually made up of or controlled by people in positions of some power seeking to depose what they see as a tyrannical, unfit monarch and replace him or her with someone more stable, a younger member of the royalty or one of its branch houses perhaps, who is frequently more easily controlled or manipulated by that very group into providing them with more extensive power. And then there's the civil revolts, which can escalate into a full-on uprising backing other members of the royalty for the throne or even a "legitimate" revolutionary group instead of the terrorist cult the Brotherhood is.

So yeah. All things considered rulers like Bellatrix require extensive use of force and/or religious manipulation to remain in power for long.