It definitely still needs friction modifier.  Auburns work using cone clutches instead of the more traditional plate clutches, but its the same material for the friction lining, so it still needs the modifier.

The only LSDs that don't need friction modifier are the gear type like a TrueTrac.

With a gear type LSD you aren't going to damage anything using friction modifier, or synthetic oil.  Its not like those things would make it not work.  However, there may be some effect on operation, likely that torque transfer would be delayed.  Probably for a tiny fraction of a second, and possibly not even enough to notice or fine unnerving.  I will note that according to Eaton friction modifier should not be used.