Blackbaud will only email a client regarding a Chargeback if the information in Sphere is not sufficient to dispute it. 
Otherwise we do not email or notify clients of Chargebacks directly. You would run the Payment Details report and add the fields: Reversal Reason and Reversal Description to identify Chargebacks to your account.
Customer Zone is used to reference the organizations point of contact(s) in cases that we require additional information to dispute a Chargeback claim on your behalf, it does not send out an automated notification. You should contact your Account Manager about updating your information in Customer Zone.