I also vote for Dino 80w90 or 85w140 if you are going to be beating on the car. In my experience, mechanical LSDs do not play well with synthetic gear oils. Almost all of the professional shops that I know of that do rebuilds recommend 85w140 in warm climates or 80w90 in cold climates. If it were mine, I'd drive it about 500 miles and drain the break in oil, followed by another drain and fill in another 5,000 or so miles.
I've read that while oil temperatures in an axle filled with synthetic tend to be cooler, the hot spots on the gears themselves tend to run higher with synthetic gear oil. Supposedly it has something to do with how the more viscous Dino lube clings to the gears and thus is more effective at removing heat.
I think you'd be hard pressed to kill the gear set with either oil and it is more about which type of oil (conventional or synthetic) is going to behave the best in regards to the posi unit.
I had a mechanical limited slip in my 07 Tacoma that would not play nice with synthetic that was rated for limited slip differentials. I put in some Lucas 75w90 that was supposed to be good for LSD and it chattered terribly, another bottle of limited slip additive did not help so I drained it and filled with Dino 80w90 and a bottle of limited slip additive and had no more issues. Of course your diff may be totally different and do just fine with synthetic.