Communication is one of the essential during Major Incident resolution. Communication should be timely and regular. During the communication (e.g. bridge call, as you suggested) you should focus on customer's business process, facts (particularly if you can explain them in some measurable way, or in monetary values) and to ensure customer that you know what you are doing i.e. that you'll bring that incident to the resolution. Also, take customers' alternatives serious. It could be that they contain some usable elements.