What LSD is it?
A viscous LSD?
Never done it myself, so I'm not sure if you can just swap the crown wheel and pinion over to the S12 ones and re assembly in the S12 case.
They are both R200's but the S13 is a short nose and the S12 a long nose.
There might be a chance of fabricated the short nose onto the S12 subframe, but then you'd need a longer prop shaft. You might get lucky and find a S13/S14 one will fit or cobble a S12 front section with and S14 rear (I know the S12 front section is longer)
You might be in the best position to see how it mounts as you have the bits, if you need a spare S12 diff I have a couple kicking around, and open 3.9 and a welded 3.7
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