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Not 100% I'm reading your post correctly. Is this right?:
(1) Your bag was damaged or lost (not just delayed).
(2) You filed a report with the airline immediately upon landing (same day as your flight)
(3) The airline later (on a different day) issued you a Baggage Irregularity Report and gave you $50. The BIR is dated with the date it was issued, not the date of your flight. It also does not state on it when you filed your report.
(4) You are trying to claim the difference between your actual loss and the $50 the airline gave you.
If that's an accurate summary, I'm wondering if you did this:
To be eligible for this benefit, you must report the loss or damage to the Common Carrier immediately as soon as you exit the Carrier. You will need to provide proof that you submitted a report to the Common Carrier, so be sure to keep a copy of the report for your records.
It's included (in bold) in the guide to benefits. Seems they want evidence you filed your report immediately, not later, in order to consider you covered.
Assuming you submitted a written report with the airline same day as your flight, and didn't keep a copy, maybe you can get the airline to give you a copy? That is an easier ask than asking them to write a letter from scratch. If you share the identity of the airline that might be helpful too.
I guess the lesson is take a cell phone pic of the written report if you plan to use this coverage.