I'm a bit confused on how much work you've done with regards to this swap. So the J30 guts went into a 240 carrier. You swapped the 4.08 ring gear to the J30 diff unit. You did not swap pinions. Somehow, the diff ends up shimmed too far in one direction. Am I about correct?
http://importnut.net/240sx-lsd-swap/
from the looks of it, you'll need the proper output shafts. J30, NA Z32s, and 240s all use different length outputs.
Double check the ring gear from the J30 has a stamp on the side that reads "47:12". You mentioned a smaller pinion, but a J30 R200 and a 240 R200 both have 12 tooth pinion gears.
The ring gear from the J30 is indeed thicker than the 240, so that would account for the shift to the right. 5-6mm is quite a bit of shimming shift though, since the shims only go up to 2.65mm.
When I did my R200V upgrade years and years ago, I landed a differential from a JDM S13 with output shafts. It was literally a direct swap. While I'm fairly certain that you only need to swap the output shafts and just ensure backlash, teeth mesh, and preload are set right in the open diff carrier, there shouldn't be too many differences between R200s to qualify such radical differences between the two.
If you can do me a favor, I'd like to know the flange thickness for where the ring gear bolts to. I'd like to compare the dimensions of the J30 diff to the S13 diff.