1987 aw11 na c52 trd lsd

You can actually have the diff out of the gearbox in about 15-20mins and only need 1 socket / wrench and a hammer to do it. (Assuming you have the gearbox out of the car and on the ground with no auxilaries).
 
Start by undoing the bell housing bolts, there are around 20 of these on the perimeter and a couple inside the bell housing near the input shaft and clutch fork (use your socket / wrench for this). Then on the diff and bell housing there are some over lapping surface points that you can tap with the hammer to begin separating the two halves. You'll get it separated till you get them about 1.5" apart and it will get a little tight, this is because the reverse gear shifter fork will be trying to come off at an angle and needs a little bit of a wriggle to get it to come off. After that, you lift bell housing off and you can see the diff, which you just lift out of the casing.
 
If it's not clear, I might take some photos / make a video this weekend on doing it.

I use an engine crane to lower the motor and gearbox onto the driveway (on top of some MDF), then put a rope across the two points and lift the back end of the car up with the crane. I have seen others do this with jacks and stands, but it's a little more time consuming.

Then wriggle the motor across the ground (slide the MDF with it makes it easier!) out into the open so you can work.
 
Working on the box in the car or just removing the box is a royal pain in the ass from my experience, so I recommend the drop motor/box method ;-)

Page 431/432 of the BGB shows about the pre-loading process that Terry commented on.

It's probably going to be a 5-6hr job depending on how good your mechanic is, so depending on his rates, it could be $500+.
Is your TRD LSD a Helical or a Clutch pack diff? If it's helical, you shouldn't find any lose in street driveability, but a clutch pack based one might clunk and chatter a bit (especially in car parks). YMMV.

No, you won't be snapping any driveshafts. Unless you have crazy amounts power and super sticky tyres and even then......