First you asked for a non LS additive gear oil and now your asking if you should add it. I'm confused. ST is not a cheap gear oil and does have an LS additive.
To answer your question, no. I have ST syn 75w-140 in all my vehicles and I like it better than the M1 75w-90 it replaced. Only one of my vehicles (F150) has limited slip.
I only asked because after I changed fluid it still makes the sound when turning from a stop.
Also my rear differential was rebuilt and I dont know what or how they rebuilt it. I look online and see pics of eaton g80 locker and mine looks different. My differential has springs when you open and look at it. The g80 locker seems more mechanical. Any pic of a 06 g80 locker?