C# Registry SetValue throws UnauthorizedAccessException

in the var for your key

var key = Microsoft.Win32.Registry.Users.OpenSubKey(Path, true);

change to

var key = Microsoft.Win32.Registry.Users.OpenSubKey(Path, RegistryKeyPermissionCheck.ReadWriteSubTree);  

Have you tried setting the accessrule and permissions?

 string user = Environment.UserDomainName + "\\" + Environment.UserName;
 RegistryAccessRule rule = new RegistryAccessRule(user,
        RegistryRights.FullControl,
        AccessControlType.Allow);        
 RegistrySecurity security = new RegistrySecurity();
 security.AddAccessRule(rule);
 var key = Microsoft.Win32.Registry.Users.OpenSubKey(subKeyPath, RegistryKeyPermissionCheck.ReadWriteSubTree, RegistryRights.FullControl);
 key.SetAccessControl(security);

One possible issue that I see with your code is that the Path variable is being set to a string that doesn't escape the \ characters. How about something like:

Path = @"S-1-5-21-1644491937-1078145449-682003330-5490\Software\Microsoft\Windows\CurrentVersion\Policies\System";