I quickly found the first issue with my PCB. I have to admit, this was a pretty stupid mistake.
Note the numbering order. It's clockwise, but should be counter-clockwise. This means pins 1 and 3 are swapped, as are pins 4 and 6. What I was thinking when I thought the numbering should be clockwise is beyond me. I assume I was looking at a drawing of the underside of the part when I was doing the pin order.
At first I thought that was it. There's no way I could fix a layout that tiny, I'd have to reorder the board. But, with some really careful cut and rewire, I managed to do the pin swapping and verify that all the new connections were good. Felt like a freaking superhero when I finished, too.
See part is in the bottom-center. It's certainly not ideal, but besides the excess flux the repair is barely visible and should hold up. Hopefully the tiny wires I had to use won't add too much resistance.
Hopefully the rest of the assembly goes more smoothly.