No. FaceContext is not a durable object. It is constructed by the FacesServlet and discarded once the FacesServlet is done processing. Since 2 servlets can't be running at the same time, you can't access FacesContext in any other servlet.
However, you don't necessarily need the FacesContext. A lot of people think that the FacesContext holds the backing beans. They're wrong. When a backing bean is in session scope, it's just like any other session-scope bean and the J2EE standard attribute access methods can be used to locate it. Likewise for request and application scopes.
Customer surveys are for companies who didn't pay proper attention to begin with.