Refresh SQL Server's service account security group membership
You can use klist purge to do this.  See How do you force an update to a user's group membership in Windows 7? and Viewing and Purging Cached Kerberos Tickets.   

I would test this a lot before doing it in production, and avoid doing it in production unless absolutely critical.  If a restart of SQL Server is completely impossible, a safer workaround would be to temporarily assign permissions to the necessary resource directly to the service account.  This would provide the access without a restart, and not require anything out of the ordinary that might cause further issues.  You would simply need to remove the temporary permission after the next service restart.