The dealer is the only place to get shims to adjust backlash and pinion depth to ensure that the gears and meshing just right.
If your differential unit is starting to struggle and you are in need of a swap, you might call Titan Motorsports. They often have gently used Limited Slip Differentials from the 93-98 Supra Turbo. The LSD from the Auto TT is a direct swap, and it would likely be cheaper than sourcing a new differential unit from the dealer.
If you already have a differential unit and just need a shop, I would try another reputable shop. If I can rebuild a differential without a shop, only hand tools, and a couple fine measuring devices, then surely you can find a shop to do the job. Once the correct shims are ordered, the differential is no more or less complicated than any another.