First.. The LSD that you have...I know where it's from, and I know its just fine.
2. Excessive Backlash can sound like a bad pinion bearing, but unless your diff was noisey BEFORE you took it in, that is NOT it. NO IFS OR BUTTS!
3. A check that you can do is this. With the parking brake on tightly and the trans in neutral, get under the rig and turn the rear driveline. There should be some movement but not much as the side/pinion gears are held tight by the clutch packs. IF there is more than 1/4" movement DO NOT DRIVE IT.
4. If #3 is correct then it should be towed to a compatent shop for repairs. I would call the shop that did the work right away and givre them first chance to correct "their" mistake, and if you don't want to do that....then the only likely way you are going to recover any expenses is to take the first shop to court.
Possible causes;
1. They just didn't measure and set the backlash correctly.
2. They didn't check the backlash in 4 spots as you are supposed to and they screwed up when mounting the ring gear to the carrier. The ring gear has high spots that are changing the backlash due to debris under the ring gear.
3. NO OIL??????
4. They somehow managed to damage the pinion when they assembled the carrier to the housing. Longshot but??????
5. ????????????????????????
If driven under load while the backlash is off "alot" the ring and pinion can be damaged and they are ~$800.00USD!

I've set up a few hundred ring and pinions so I speak from experience, and I've taught professionals in this industry to do this also.
Sorry, but again...more good news.
Kary