So yes, I've worked full time on SHOs for the last 17 years, and have rebuilt many trannies, including Quaife and Wavetrac installs. A big advantage on the track, but overkill for the street as it's rare to be able to go round a corner fast enough to have them kick in. But they do overcome the diff pin problem. The Quaife went south a long time ago. Doesn't look like the Wavetrac is available, either. As an aside, it seems like the Wavetrac generates a lot less heat than the Quaife (per my experiences with the Team Dick Lemons car), to the point where a transmission cooler may not be required, unlike the Quaife (per my track car).
First off, I'd give the Traction Concepts LSD a wide berth. If you look at its pic and that of the Phantom, they are identical. The Phantom has already proved to be a piece of crap.
Second, the shop manual does an excellent job of transmission assembly / disassembly. OG SHO Guy sent you some pages from the manual, but didn't cover the assembly / disassembly part. Josh's video should cover that. When you review it, I think that you will find that you had the tranny the 'wrong' way up when you took the case half off. It's normal to have the bell housing at the bottom, leaving the diff and shafts in the bell housing. Then it's a matter of grabbing the input, output and gear selector shafts all together and lifting them out as one.
Third, what might be causing the case halves not to mesh properly may be the meshlock plunger. This is a roller and spring at the top end of the gear selector shaft (assuming that the bell housing is down and transaxle case up). The roller interferes with the transaxle case, and has to be pushed in with a small punch to clear the case.