If you are truly pissed then do like I mentioned above and file a claim with BBB. I would also file one for the shop that did the original install!!!!
I think the majority of the fault here falls onto the installer shoving the wrong parts into the diff and letting you drive off like that. Sounds rather shady to me and that is where I would focus my attention, it's actually very clean cut. It happens all the time where people bring their own parts to auto repair shops and it is absolutely that shops responsibility to ensure they are the right parts. If it needed as much shimming as you say it did, there should have been a ton of red flags.
Now if the LSD exploded after a proper install then I get it... And I see why Randy's won't refund you, they didn't do the install. I am sure they would if your shop would have noticed, and you wanted to return un-used parts. I will agree that yes it is bad on you that they identified it as working for your car, but I also bet in there terms of service agreement they also state that at the end of the day it is your responsibility to know what goes on your car, not theirs. Is it bad customer service to have that info on their site, yes. But it's not then end all be all if you ask me.
File a claim on the crappy diff shop that did it the first time, you have a chance to win that one and get the parts and labor refunded in full!!! Also do what was mentioned above with the state, especially with a local shop!
TLAD: Don't always believe what is on the online part stores websites