This is not an answer but a followup question. In the p-adic case, is there any hope for a a set of conditions on the local Langlands correspondence which would make it unique? In the case of GL(n)$GL(n)$ this is provided by L and epsilon factors. For classical groups, can one make a precise statement (even conjecturally) for classical groups, using lifting to GL(n)$GL(n)$?