Thought I'd update this post as I have still not solved my inside tire spin issue. I went to VIR a few weeks ago and had Quantum Speed Works look at the suspension. They could not find anything that should have been causing the wheel spin in the suspension itself. They felt that the diff was not locking up.
So, I decided to upgrade yet again. This time, I'm stuffing a 6-disc conversion in there. I went with RacingDiffs Stage-3 package: https://www.racingdiffs.com/collections/lsd-clutch-plate/products/bmw-188-mm-lsd-clutch-packs?variant=33470318739591
VERY IMPORTANT IF YOU ORDER: They need to know what your final drive ratio is. Email it to them after you order as they request it on the product page but there's no place to enter it when ordering.
It's a bit of a CF trying to communicate with those guys. They're responsive and they try to be helpful but their English is only marginally better than my Bosnian.
The biggest issue I have is finding good descriptions of how the parts are supposed to go into the case.
I found a TIS guide on LSD disassembly/assembly here, but it is for the stock 2-disc LSD:
https://www.newtis.info/tisv2/a/en/e36-323i-cou/repair-manuals/33-rear-axle-rear-wheel-drive-guide/33-14-locking-differential-with-bearings/BIHpnmD
I'm praying that I'm understanding the orientation of the conical spring washers (#3 Disc spring).
RacingDiffs doesn't use the same nomenclature as is found in the PDF above, so the mapping for the parts that are replaced is:
BMW Nomenclature ------RD Nomenclature
________________________________________________
1 Case Covers-----------Billet Steel Cap
2 Thrust washer---------N/A
3 Disc spring------------N/A
4 Tab washer------------Oil shim set
5 Shim------------------N/A (4mm OEM shim not used)
6 Disc spring------------Preload shim set (Belville Washer or Spring Washer)
7 Outer disc-------------Part of 6 clutch set - dog ear plates
8 Inner disc-------------Part of 6 clutch set - friction discs
9 Thrust ring------------N/A - this would be what RD calls a Ramp
10 Differential side gear--N/A
11 Differential pinions
with differential axle--N/A
12 Expansion housing---N/A
The parts they list as "6 clutch set" are 6 dog ear plates (#7) and 6 friction discs (#8). The RD dog ear plates are coated with friction material (Molybdenum-Ceramic coated) as well as the friction discs. Their oil shims (#4) are also coated.
The one thing that RD includes that is not OEM is the Ramp Saver Shim (basically another dog ear plate but it is not coated with friction material). This is only necessary if your Thrust Rings (Ramps) are scorched and need to be surface ground. If they do need this, they recommend taking them to a machine shop for the grinding. Have them take off .5mm (I assume .25mm from each). The you will add back that space with the Ramp Saver Shim. They provide a PDF when you order that explains this and where the shim goes in the stack.
The main differences from what's in the PDF above is that you don't reuse your old #5 shim (the 4mm thick flat shim) and there are 3 Inner (#8) and 3 Outer discs (#7) on each side of the Thrust Rings. You'll alternate these so against each Thrust Ring you'll have inner, outer, inner, outer, inner, outer. Everything on each end of this stack should be the same as OEM. (NOTE that RD does not provide any instructions. I had to email them to find out what order to put the parts in)
One other thing is that you have to use their Case Cover (Billet Steel Cap) because it has a shorter shoulder than the OEM cover to make room. That means you either have to press off your old bearing and put it on the new cover or you have to order a new bearing. RD offers a kit with all the bearings and seals but it's 165 Euros. I could not find the part number for the bearing anywhere. I emailed RD and they graciously gave me the part # and I picked it up for $26 from Pelican Parts: BMW Carrier Bearing - Differential Output Shaft - 33131204568 - FAG 103133
I chose to order a new bearing because I was afraid that pressing off the old bearing could damage it. Installation should be easier. I found a YT video where the case cover was put in the freezer and the bearing was heated with a propane torch. Then the bearing slipped on easily. So I'll give that a try tomorrow.