I've had to submit a claim like that sometime ago, I simply submitted a claim to AFKL for a confirmation of no reimbursement on their end stating I had insurance with AMEX, which they were happy to confirm.
I was able to avoid waiting on a possible refund from them and go straight for AXA claim, while meeting their requirement of no refund with the airline.
As to if that's contractually valid, I couldn't possibly say