First and most important, you should probably take this post down right now. Whether or not you have a confidentiality agreement of any sort with this company, saying bad things about an "anonymous" major player--which could be just one of two companies--exposes you to potential liability and a headache that you do not want to deal with. I put no research into this, it's just my immediate opinion.
Second, ordinarily you cannot recover punitive damages for a breach of contract. You can recover actual damages and sometimes (rarely) attorneys' fees and costs.
Third, the size of the law firm is proportional only to the fees you will be charged--fees that you probably will not be able to recover. More important that office size and advertising budget, make sure that you get along with your lawyer, trust him or her, and be certain that the lawyer has adequate resources to handle your dispute--which sounds pretty basic. You do not need a firm capable of litigating a federal antitrust prosecution or a multi-party international-trade dispute. Focus on the lawyer, not the brand.
Good luck.