It turns out that the set Voltage (!) is what makes the issue appear or disappear as shown on the video below.
Basically, if the set voltage is below a small multiplier (3-4 in this case) of the burden voltage, it's all fine. But if the set voltage is higher, the burden voltage increases and so does the current until current reaches 1.1x, then both stabilize. In the video, I go up then back down in voltage. As the set voltage crosses 2.5-3V, the transition occurs.
The top DMM is showing actual voltage (matches the power supply's voltage readout and the bottom DMM is showing current).