These cases are not so easy to win. I would have to read their terms and conditions as they likely have a disclaimer. They will say you are "gaming" the system or "reverse engineering," in an unethcal way. You might also be required to divulge your "secret" for getting the likes. You sound very bright.
At this point I would need to read the T&Cs and anything on their site, then write a firm letter on your behalf. Of course, you need to consider what it will take to sue them ( a lot ) or ask for a settlement. You can also put your complaint on websites to discredit them, as long as you're telling the truth.
I am quite good at writing persuasive letters, but the key to a good letter is affirming the law, which takes time to do. I would never write a letter without knowing how strong your legal position is.