Well... it lasted a few weeks and crapped out again.
This time, when I probed the CLK pin I found there was no clock signal when commands were sent to the IC. In fact, there was a lot of noise on that line indicating that it was probably floating. This was not the case last time I was troubleshooting this, so it must be intermittent and just happened to be working again when I was measuring it the last time.
So I visually traced the CLK signal on the PCB (thankfully, only a double-sided board) all the way back to the MCU. Then I measured from one IC to the other with my DMM and... open circuit, and occasionally 20-30 ohms if I gently twist the board a certain way.
Eventually I narrowed down the problem to a bad via, as I have near zero ohms on one side of the via, and high resistance on the other side of the via. To repair, I removed some solder mask from both sides of the via and a bit from the trace, threaded a strand of fine wire through the via, and soldered it down to the exposed copper on both side. It works!!! Hopefully for good this time.