A couple of things are unclear from your post: (1) Did you/your friend try to resolve the problem with the hotel prior to challenging the charge on the credit card?, (2) Did your friend dispute the entire charge or just the dollar amount of the duplicate charge?. It doesn't make sense to me that any business would write a" 9-page letter" to the credit card company saying you need to pay. Perhaps a 1-2 page letter with supporting documents? If you didn't contact the hotel first to explain the situation, they may think you are the scammer.
The best strategy is always to deal directly with the business first. At this point, I would write a polite, non-accusatory, brief letter to the GM. Write an opening sentence or two about wanting to correct a mistake on your bill. If this is your first contact with the hotel, you may want to apologize for not reaching out to the hotel before going through the credit card company.
Then bullet point the facts in a new paragraph of 4-6 to sentences: "Mrs. Smith reserved x rooms for x nights at $x/night for her family and my family on xx/xx/20xx. On arrival on xx/xx/20xx, I provided a credit card to pay for my family's 2 rooms, and Mrs Smith used the card used for reservation to pay for her 2 rooms. The total charges for x rooms and meals were $X. Her card was billed $X; my card was billed $X. It appears that we were double charged for my 2 rooms."
Then state what you want in a new paragraph. "I would like for Mrs. Smith's credit card to be credited the cost of our rooms, which is $X. Thank your for your prompt attention to this matter. I look forward to hearing from you soon."
Attach supporting documents, like the reservation and the folios for you and your friend. They may expect the letter to come from your friend if the money is going back on her card.
I would strongly recommend against social media threats, especially if this is your first interaction with the hotel. I would strongly recommend against charging the hotel with scamming you. I would strongly recommend against subtle and not-so-subtle digs like "The total charges for our substandard rooms and terrible meals were $X." Those sorts of things tend to make a business dig in its heels. Be dispassionate. Be courteous. State facts, not opinions. Also... be brief. No one wants to read 3 pages about this.