This isn't what credit card chargebacks are for, since the merchant in this case did absolutely nothing wrong (and even did more than they were required to). Next time, be more responsible about booking stuff like this and pay extra for either travel insurance or a flexible cancellation policy.

All that said, you can try to chargeback and you might get lucky, but keep in mind banks can and do drop customers who abuse chargebacks. If this is your first one that won't be an issue however, even though it isn't legitimate.