If that is your concern, there is no other realistic way than to sort it out with Emirates.

The "ban" that you received is probably a more or less automated follow-up to any chargeback and does not involve any (re-) review of the underlying issue with your flights.

Even if it goes against how you feel about the situation, your best chance to resolve it with them is to contact them, be a bit humble, and focus exclusively on the facts of the cancellation and refund situation, i.e. no moaning about how bad their service is, how this is all their fault, how you gave them enough time to solve it, etc. Simply state that you would like to resolve the situation, that the chargeback was the results of a misunderstanding and ask them to recalculate the fare based on the cancellation, then let you know whether (they think) you still owe them money.