There is no reason why this should not work. For example, at my current contact center project the agents have two lines, a CC line and a personal line. When they log in with CAD they specify the CC line. The device target and the label is only on this line - naturally.
As an aside, you should set the registry of the PG to disallow logins on JTAPI associated devices where the device target is not configured.
HKEY_LOCAL_MACHINE\SOFTWARE\Cisco Systems, Inc.\ICM\customer\PG2A\PG\CurrentVersion\PIMS\pim1\EAGENTData\Config\RejectLoginWithoutDeviceTarget
Set to 1. This will prevent agents from accidentally logging in with their private line.
CAD will have no trouble using the extension the agent logged in with. Are you sure they are logging in with their CC line?
If you are having trouble placing a call, it could be your CUCM config for these lines. Check the Calling Search Space.
Regards,
Geoff