Taking notes from my previous attempt, hastily cobbled together:
It used a quad jeft op amp, a boost board for ~6v, then a 5v and 3v ldo for power rails. The 1khz noise persists. Looking at capacitance multipliers to filter out the rail noise, though I fear the noise is from an internal regulator.
I'm not sure why, but the noise source wasn't apparent when I was testing on a 4.x linux kernel. Only windows and 3.x kernels.