Cone style suregrip synthetic gear lube.

My thought is this...
I had a good friend who ran his own rear end shop. He specialized in 9in fords for racers, but would work on anything. He did my Detroit locker 8 3/4.
He refused to use synthetic in anything. If the customer insisted on synthetic gear lube, his policy was "warranteed for six seconds or six feet, whichever comes first."
He used Torco in everything.
Me too.
Edit: if you don't want to change the lube, mitch's ( my buddy) advice in your spot was to find an empty parking lot (shouldn't be too hard, now) and Do a series of slow, tight figure eights, to work the lube well into the cone.