First, CALEA requires that the company provide a "senior official" as a point of access for law enforcement, and a backup 24/7 contact. Their senior official is their general counsel, and the backup is another lawyer. It's not their network operations center. Their general counsel wants to see a warrant, and checks back with the court to make sure it's valid. This is the way to do it; bring in Legal.
There's a procedure for "emergency requests" in advance of a warrant under CALEA. This telco immediately faxes the law enforcement requester a brief form to fill in for those. It requires name, police department, office address, badge number, and a brief explanation of why there's no warrant yet. The form to be signed by the law enforcement official contains a statement that the official will provide a warrant within 48 hours, and in the event that they fail to do so, their department will take full responsibility for their actions, including indemnifying the telco against any costs and damages. That's followed by a statement that in the event the law enforcement organization fails to authorize the actions of their official, the official will be personally responsible for said costs. The telco also reserves the right to disclose requests for which a court order does not follow.
This discourages fake "emergency" requests. Get your legal people to draft something like that. The key to this is that law enforcement's interface to your company should go through your legal department.
(I used to have a link for this, including their forms, but can't find it now. Can anyone else find it? )