I don't know if it matters, but how do you use the card? Swipe, chip, tap, via Apple Pay, something else? If it's something like Apple Pay or the Google equivalent, maybe your card info got corrupted somehow. Removing and re-adding the card might fix it.

You mentioned travel - was that big purchase close to home or in a location you often travel to? If it's your first time in a new place making a big purchase, maybe something in their fraud algorithm flagged that (which makes sense), and each time you make a new purchase, their algorithm is seeing that transaction (again) and re-flagging off of it. Not sure how you'd fix that, but if you're calling during US business hours, consider insisting on escalating and staying on the line until a supervisor becomes available. Simply put, you have to find a way to make it Amex's problem, not just your problem.

Finally, calling out Amex on social media for their failure to call you back (NOT for their fraud detection) might work in catching the attention of a bigwig, or at least someone who can contact a bigwig for you. Focus on the poor customer service (repeated failures to call you back, ineffective supervisors), and not on the fact that your transactions are getting flagged for fraud. Social media monitors can help with customer service, but they're likely to stay away from fraud discussions just as a security thing.