Another update!
Recently I got the chance to experiment with this a bit more and managed to make further improvements:
First, I removed the 100K resistors from the latter stages' non-inverting inputs, as these resistors were not contributing in any significant sense.
Second, and the most interesting to me, is that I changed the second amplifier stage to non-inverting. This further flattened out the noise spectrum. I'm unsure exactly why this is.
Third, I AC-coupled the input of the first stage. This, along with the first stage's matched input resistances let me DC-couple its output to the non-inverting second stage, as the DC offset is negligible.
Version 3 schematic and scope shot attached.