been a while since i posted here, but here goes.
firstly, i think the dealer is screwing you. that work isn't cheap, but its nowhere near that expensive.
i drive an mx83 cressida up in canada so we have an lsd in them from factory virtually the same as yours. i've been having the same problems and its quite fixable but theres a lot of work involved. what i would do is this:
first, find yourself a spare diff and swap it. by far easiest and most cost effective way of solving this. i did that but turns out my spare is a stinker too. for that reason i decided to fix my old one. i;m guessing you're getting a whirring noise that sounds like its from the back of the car or even near the trans area. as the bearings are trashed they can create a vibration that can travel up the pinion shaft and resonate in the transmission. don't worry, it shouldn't cause any damage unless you count pulling your hair out in annoyance.
if you love your supra and you want to fix your diff, still get the sapre if you intend to drive it over the next couple months. its a lot of work to fix your lsd.
since its bearings, i won't get into if you have bad gearing or bad clutch plates as that doesn't seem to be the issue. all bearings excluding the outer pinion bearing must be purchased through the dealer as well as new oil seals, snap rings and a crush sleeve.
you don't need all the SST's but disassembly will be a bit different than what the toyota factory manual describes. you will also need access to a hydraulic press to get bearings on and off as they're an interference fit. other than that and your usual set of sockets/johnson bar, you'll need the following:
impact wrench (to get pinion flange nut off - its a bitch)
a good length of chain (bolt on drive pinions must fit through the loop, you use this to yank them out)
nuts and washers to fit those bolts (can't remember size, but they're a metric fine thread)
strong set of gear pullers to get bearings off (if you don't want to reuse good bearings)
a hydraulic press and a bearing spearator if you do want to reuse any bearings...
good click type torque wrench, the usual 200-250 ft.lbs max range.
a small torque wrench with a range down to 0 in-lbs and capable of measuring into the single digits. small beam type is best, very hard to find.
metric or imperial micrometer with thimble
dial indicator and magnetic base holder
dead blow hammer
best place i've found to get toyota parts would be
www.1sttoyotaparts.com. i saved over $100 on bearings even with shipping!
be warned, a repair such as this is not the easiest job. i'm doing it because i want to learn and i understand the risks. theres a lot of hours involved with it. i don't know what your skill level with repairs is at, so fair warning
if you're currious i've been slowly building up a guide for this job over at
www.toyotacressida.net (you have to sign up to see the real forums). its more of a worklog at the moment, but you will probably find it useful.
finally, follow the mk3 TSRM for more info. like i said, dissassembly is slightly different without the SST's but is very doable.
hope that helps.