Another problem with filing the dispute is that you now have two avenues in process with different parts of the company. Escalate with the executive team first and if that doesn't resolve the problem, contact your credit card company. Others are correct that disputing the later charge is just flat-out wrong (the wrong amount as well as the wrong tactic). It was a direct purchase by the cardholder with the airline and would punish the airline for a refund owed by someone else entirely. That is morally wrong in addition to opening a bigger can of worms of creating more work for everyone involved, including the OP, for absolutely no reason as it will be denied. Some vendors will blacklist you if you do chargebacks that are not warranted.
The credit card wants the customer to have worked with the vendor to resolve the issue before getting them involved. In this case, the OP has done that and is already being promised a refund but needs someone to push it through on X's end, which will likely happen with contact with the executive team.