I would make the claim to CIti and also send a note to AA customer service in which you say that you are making a claim to Citi and will follow up if you are not covered. If the Citi claim takes a couple of weeks and is denied for whatever reason, you will be in a tough position making a claim to AA that far out.
Even if CIti does not pay, your standard travel insurance may cover this even though that's not why you purchased it.
There is no reason for AA not to pay the fair market value of the item, cheap or expensive. The key is "fair market value." A jacket costing $500 worn 10 times does not have a FMV of $500.