FWIW, the difference between First and Business isn't that great these days. Was it really worth cancelling the whole trip just because of that? As others have said, some routes don't have First.

Regardless, you need to wait longer for a refund. Five days is not enough time, and filing a chargeback may only delay matters more. I've had other vendors who owed me a refund say explicitly not to file a chargeback while they process the refund. Once filed, they put refunds on hold while the chargeback is investigated to avoid refunding and getting a chargeback at the same time.