For limited slip differentials, it's usually recommended to drive the vehicle in several tight figure of 8 patterns after changing the lube to get the new stuff worked in.

If it continues to chatter after that, then I'd say you would need additional external friction modifier added. Just enough though, too much will render the limited slip useless, or ineffective.

Most synthetic gear lubes on the market now, already contain the friction modifiers in their formulation. This is one area where I feel synthetics are of benefit, compared to the crankcase.