I would take the lsd out of the old rear, swap the ring gear from the non lsd onto the lsd......reuse the shims with the gears and housing all same regardless of lsd.
basically what i'm trying to say is keep the gears with the same housing using the shims with that gear and housing and the lsd should bolt in just fine and have the same lash as long as these gears are used in the same housing with the same shims.
Get some gear pattern paint or grease whatever they call it and double check the pattern and you should be all good.
Here is some info on doing a diff, I didn't really read it but it had some figures and looked decent for the quick browsing I did of it.
diff articleHere is a decent figure to show you contact pattern.