I will leave the decision to you, obviously, but it might be worthwhile to share this thread with your 'senior' person at AEXP Travel so they know not only what is going on, but what we think about their capabilities. I've had my own issues with rebooking, which like the OP, required email and phone blasting everyone from Manila to Phoenix until an actual supervisor called me back while sitting in front of his Sabre session and I spoon fed him exactly what I wanted, he confirmed with the airline that my request aligned with their rebooking policy at the time, and the changes were made. Why did it take so many phone calls, chat sessions, emails and hours spent to do something that should have taken 20-30 minutes including phone time with LH? Who knows.

For the annual fee I pay for the Platinum card, I would expect ONE phone call to a US based travel agent (NOT a customer service script reader) who is empowered to take care of the problem in whatever way is possible per the booking rules and rebooking policy of the ticketing carrier.

I know I have another fight coming up. I have a refundable-for-a-fee (in TWD to make it more complicated) ticket on BR that I already upgraded, and that thanks to my in-laws I will likely need to cancel and rebook, and I already had THREE arguments with Amex agents insisting that the ticket, at time of purchase, was non-refundable even though I spoon fed them the actual text from the voluntary change/cancel rule in the tariff display. They finally relented and confirmed it was refundable before I bought it, but now that I probably actually need to do the refund, getting them to follow through and calculate the USD equivalent penalty from the tariff rule is going to be a nightmare, so I will need to save some of stiff drink for that fight to come.