Hi --- It's amazing what weight a card co like Amex could bring to bear in securing your refund, once you have their financial muscle on your side you will find things get done.
If I was in your position I would involve them right now by asking for your money returned by them under the consumer credit act, they will then pursue the company with lots more pressure than you could possibly apply.
You have tried your utmost .... Maybe it's time to take the gloves off. !!!