The normal 90 weight "multi-purpose" gear lube would be fine for that rear end. It's got the hypoid additive needed for proper lubrication of the ring and pinion that the "mineral oil" lube for transmissions doesn't have. The cone clutch limited slip rear ends have a bad shudder when the additive isn't used. If yours is just making a rapid click, it will do the same thing with the additive, also. The difference is that the Detroit Locker actually mechanically joins the axles together- - - -it doesn't use clutches. Another place the locker was used was in the GT-350 Hertz Mustangs, and it really spooked some of the people who rented those cars.
Jerry