That was it! I lifted pins 3 and 5 and tied them to pins 4 and 6 of the data selector. The display is bright and working now, meaning the shift register config register of the FM6126 was just not getting programmed until I could toggle the clock while the latch was high.
One problem remains: the last vertical column of the display looks like it might be showing address lines. I’m not entirely convinced the whole display isn’t just shifted left by one pixel… but I can’t verify that yet… I suspect you had that selector there for a reason, I just don’t know what it is yet. 