AMEX is probably the very best internet bank.
For some reason, they are suspect of the check and (or your account history). They have not accused you of fraud...they merely have informed you of the possible consequences.
Yes, it will be an inconvenience for you in clearing this up. You did not give details as to how you submitted the check... but believe me, AMEX interest is not in declining a deposit. They have an honest security concern.
Sometimes AMEX makes me double confirm a transaction... a slight inconvenience.. but one I actually like as they are exercising good security.
In case you are unaware, there is a fair degree of fraud concerning government checks...(I don't think the government has any great reconciliation and confirmation procedures).