Um... It works... both my S14's have been running the same J30 LSD for years. Only issue is that I had to swap out the axles because I was too lazy to swap the output shafts on the diff for a set of 6 bolt output shafts that would work with the stock open diff's 6 bolt axles.
No, you don't need to swap out the back diff cover for a different one, just leave the J30 diff how you found it and bolt it up in there. You won't need to hack the sub frame. And it won't cause the drive shaft to be wedged in at all, the J30 diff and the stock open diff are the same in all the important dimensions that effect fitment. Just do it, and enjoy the throttle steer.