I had a similar experience with this kind of claim, it was not via Amex though. It was through a Visa card. The card insurance covers up to $10000 for purchase protection. But they want to be secondary (to home insurance). So I made my deductible of my home insurance to $10000 so I won't be forced to file a claim against my home insurance for every purchase protection claim.

I purchased a piece of jewelry worth $20K+ and lost it in on a Europe trip.

We got a police report in Europe (Hungary or Czech republic, don't remember) The card company made me file a claim via my home insurance. They covered up to a max of $5K for jewelry with a $10K deductible. The process was pretty simple. No questions asked. I got a check for $5K from my home insurance and $10K from the card company insurance. It seemed no big deal to them.

I would assume Amex would be the same. If you get your police report, they really can't contest it.