Statements for my International Dollar Card are mailed from Fort Lauderdale, Florida. The Euro card may be different. Of course there is no need to wait for the paper statement to mail the payment, all the transactions are online.

In my opinion the "due" date shown on an American Express charge card is a fiction, the payment just needs to reach them prior to the next closing date.

I am able to pay my International Dollar Card using online billpay from a US bank. I am not confident this would work with a card denominated in another currency, but you could try submitting a small payment to test if it is applied to the account, how long it takes, the exchange rate applied, and if they assess any fee for a payment received in USD rather than Euros.