Undertaking a claim that there is no one you can speak to Facebook about this will help them quickly find someone to talk to you and respond to the credit card company and you.
When standard channels aren't working, complain to someone who can do something about it.
A couple of times in my life, I've had good luck contacting the office of Chairpman of the Board. I had problem with a debt collector of American Express, I was a financial delinquent in those days, and I contacted American Express directly. I asked for the Chairman of the Board's office, and talked to his assistant. I explained the situation about how I was being lied to by the debt collector, the assistant arranged to have the debt recalled to American Express, and I worked w/ Amex directly after that.
Another time, also w/ Amex, my friend's consulting invoice wasn't being paid my Amex in a timely manner. I recommended she contact them using the same approach, and an assistant helped her get paid.
More recently, I had a problem getting my tax return processed with the Internal Revenue Service. I complained to my Congressman, his office put me in touch with a tax advocate at the IRS, and everything was cleared up. This is the second time I needed to do this.
Yeah, disputing the charge might be better than a straight chargeback. The merchant gets to respond to the dispute and provide evidence, and if they clearly deny having issued the charge then it gets reversed and you can move on.