Unfortunately, this isn't an option. Badge earners can choose to make badges public or private and can also decide to share their badges to social media (or not). Earners have full access to visibility controls and must take action to share badges if desired.
A badge is a digital representation of an earned credential/certification, and for verification purposes, earners are not able to change the information stored within a badge. The issuer controls that information. The earner's full name is used and displayed for verification purposes. Email addresses are never displayed and, earners are in control of how much is public or private, through their profile "Account Settings".
At this time, Credly does not have the ability to allow earners to display their badge without displaying the issued to name. If the badge is public, the name displays as part of the badge details, when a badge is set as private, only the earner would see those details.