Issueing a chargeback is not always as simple “exercising you right.” You get your money back, ok. But you may still owe the provider the amount that you have taken back - or the goods received. The credit card provider is not legally entitled to make a decision on that matter. So the store owner is entirely entitled to refrain from servicing you until you pay up.
The problem is that I have issued a chargeback by mistake (I confused the charge from the store with something else, as they have changed the name in credit card reports). I immediately called my bank back and explained that this was a mistake, but it was too late — the chargeback processed, the card reissued, my last order in the store marked as “fraudulent”, and I am banned for life from this store. I even cannot pay again for that order, as they are not accepting anything from me, and their support is telling me there is nothing they can do.
Have they banned you for life, or just that card? I can't really see how they (especially assuming not a tiny low-volume owner-run business) could do the former.