The AMEX car rental coverage is pretty weak, and the reason I continue to maintain a Diners Club account. First of all, the AMEX coverage is secondary, so if you own a car you must submit the claim under your own car insurance coverage first, and AMEX will only pay anything unpaid (i.e. your deductible). Obviously this risks boosting the rates on your personal insurance. Second, unless AMEX has changed things, this "pay for the entire rental" rule is much stricter than Diner's. Diners allows you to use coupons as long as there remains a charge for at least one day which is billed to your Diners. So if you have a free weekend day coupon and use it on a two day rental with Diners you are still covered. Rent for only one day and use the coupon and like AMEX there is no coverage.
As far as your current problem, it sounds like they are correctly denying coverage under the terms of the program. Best I can suggest is getting the full terms and conditions and seeing if there is some exception or technicality you can shape your claim to fit under. Nothing comes to mind though.