Tried separating the power to the sensor circuit board from the driver circuit board (so the only connection they share is ground). Seemed to work at first:

(The messiness in the first half is before I ground connected other set of power supplies to earth ground).

Close up:

But I tried to insulate it (by wrapping the padded aluminum around it), and then the voltage drop started to show up again (and remained after I took the padding off):

I used to oscilloscope to monitor the power voltage signal while entering commands, and none of them dipped or showed any disturbances.

Andrew suggested that the sallen-key filter on the sensor board could be causing the drop, so I disconnected the filter, but it didn't help. However, as I was using the oscilloscope to probe the driving signal (and sensor signal) at different points, I found that having the oscilloscope parallel to the heater (at the leads) caused the voltage drop to stop (tentatively, only two pulses to go off of):

(Also, noticed loading effects when probing along the signal line)

I thought it might be possible that the metal slab might be adding some stray capacitance or otherwise causing interference between the driving and signal lines, since, other than ground, it's the only place where the two circuits are really close to each other, and that the impedance of the oscilloscope changed the "impedance" of the slab. So I thought I would try to use a 1Mohm resistor in place of the oscilloscope, but when I was trying to attach a 1Mohm resistor, one of the leads to the heater fell off :

I'm not sure how the best way to go about fixing this would be. I think maybe just clamping the leads to the heater for now, since soldering this well would be really hard...