Bit of an update, I'm still trying to find that vibration and am going to pull the rear end apart this winter. It's got to be something in there, it's just got to be. I'm suspecting something to do with the pinion bearings, but I'm not entirely sure. Recently I remembered that when we were assembling the differential with the TRD LSD, I had also bought a TRD solid pinion spacer and shims, but we were unable to use it because the spacer was actually too tall to fit. We ended up just using a crush sleeve instead. Thinking about it I am now wondering if there were actual bearing issues that we didn't realize or if I got the wrong bearings in the box, or I don't know, but I need to find out.
Also, another small update is that I found what I think is going to make the front end of the car look a lot better. I have had a new kouki Trueno emblem sitting in the package for years and years but never wanted to use it because my car is completely badged as a Corolla GT-S. I wanted to find something that would work and look good but say "Toyota" instead of "Trueno", well unfortunately I'm not sure exactly what I want exists from Toyota and I am looking into other options (like 3D printing something), but in the meantime this was the closest I could find, and I think it should look pretty darn good, even with though it's a chrome emblem. (Comparison beside a zenki emblem.)
I haven't mounted it yet, but I will soon I hope.
Jeff