Force MS SQL Service Account to re-authenticate
When a Windows account logs on to the domain the session receives a ticket that includes all of the group memberships.  Accordingly, if you add the account to a new group, it will not be in effect for sessions that are already logged in.  So you'll need to either:


Restart the SQL Server service
Add the user account specifically to the share permissions as a temporary solution until the next time the SQL Server service is restarted
Change the path of the backup config to use an IP address instead of a servername--again, temporarily.  This will cause the connection to bypass Kerberos and use NTLM authentication, which will pick up the group membership (not 100% sure this will work for you, but it's totally a thing).