L.s.d

[Conkers]

Strewth.... it doesn't matter if it is a live axle or not,

First measure the backlash on your s12, (backlash is how much the crownwheel rocks on the pinion. Keep a note of this as you are going to use the original crownwheel again on your lsd.
Then you remove the guts from the kingcab rear axle, remove the crown wheel and bearings and throw away everything except the diff cage (with plate type lsd).
Then.. as long as you have the same bolt diamters, (apparently some were 10mm and some 12mm) fit your original crownwheel from your original non lsd (or gay viscous) rear axle onto the kingcab plate type lsd guts.
Fit the bearings, check bearing preload, check the backlash, moving the shims around to get the same backlash as you ran with your standard diff, otherwise it will whine...

And bobs your uncle...

Conkers <--- thats all I got to say about that.