Well, the MCU was the culprit. The problem with being stubborn and obsessive is you stay up past your bedtime and do things after thinking about them some more, because you just can't stop thinking about it.
So I reflowed the MCU with a new one after breaking out my hot air rework station which was tucked away. Now everything works fine. And, to make things better, I found two things. One is a bug in my code (well, a line of code I failed to remove), and the other is the fact that I think the power supply on the PCB and the power supply in the programmer like to fight each other. Not sure why, but from now on, when programming the MCU, I'm going to turn off the input power supply. One of these two things might have caused the issue.
So, now I can verifiably set a voltage (say +4V, or -8V) and have the MCU set the voltage at the DAC such that the output goes there.
One thing I've found though is that my center reference voltage change from 1.65 to 1.0085 has screwed up the DAC output, in that the full voltage swing isn't covering the entire capability of the IC now. I can go full negative, but not full positive. Full positive is now limited to +9V. This is an easy fix at the gain stage level really. It just means instead of having +/- 30mV resolution, I will have +/- 50mV resolution. The fix is to increase R9, or decrease R8. Right now I've got a gain of +10, so 0V input, 1V reference = 10V out, and assuming 1V gate-source, that's about 9V max. I want ~15V out, so I'll need a gain of 15. This means R8 to 3.3kOhms, or R9 to 75k (likely I'll go with the latter, as I use 75k resistors elsewhere).
I will make a video tomorrow at lunch I guess!