Since it looks like AM at the input side is not the root of our current low-frequency problem, we have been trying yet again to come up with new sources. Alastair had the idea that pointing noise of the beam going into the cavity might be causing us a problem in some not-yet-well-understood way (this was first discussed in the context of 19 MHz jitter from the EOM being converted to RFAM as it could be in the AOM, but the cavity should have very little response at 19 MHz with which to do this).
I decided to look at the low-frequency spectrum of the DC_TRANS signal to see if there was the same type of behavior as in the gyro spectrum. As it turns out, there is. Below is a plot of the two compared side-by-side (though I had to scale the DC_TRANS plot up by a factor of ~10 to get it to coincide). There is extremely good agreement in the shape of the curves, and I think we'll see that the HF floor of the DC_TRANS plot is just the noise floor of the (broadband wide-area Thorlabs) PD.
I can't say that I can think of any obvious coupling mechanism here. I think it's feasible that pointing drift of the input beam is causing the spatial eigenmodes of the cavity to wander differentially (i.e., the points at which the supported mode in one direction touches the mirrors all move horizontally across the mirrors a bit, making the whole square 'rotate'---this is dependent on the input beam and can thus happen independently in each direction). In this case, the cavity length would differ in the two directions, resulting in a gyro signal.
We need to do some more work to figure out what exactly is going on, but this is another data point that helps with the diagnosis. I think the next step is to look at the power at some pickoff point close to the laser to see if this is an input power drift or something caused by varying degree of coupling into the cavity (as from misalignment).