Your payment form only allows for a three-digit CVC. But American Express cards require a four-digit CVC.
I was able to successfully pay though by inspecting the HTML and editing the form.
Change from:
<input class="form-control" data-stripe="cvc" id="card-cvc" maxlength="3" size="6">
to:
<input class="form-control" data-stripe="cvc" id="card-cvc" maxlength="4" size="6">