Today I'd decided to take another look at this. Close inspection revealed that the spring that provided the tension to flip the mirror up and down had broken at the point where it connected to the shaft. Bending the end of the spring a bit and inserting it back into the anchor point fixed it.
A few more adjustments to add more tension got the mirror working back to normal. Adding more tension also solved the problem with the mirror previously sticking occasionally at speeds under 1/60.
Now I need to load it up with film and get out shooting!
If you want parts for this, I have a body that didn't work that I took apart (so it's essentially a pile of parts). But you can have it, if you want to give me a couple bucks to ship it.