With the pump open at the max power of 10W (after the chopper), I noticed that the control loop introduces a noise at a constant phase.

Without the loop, the cloud of points is 40microV large (radius) and centered at 20microV (phase around 150deg)
With the loop, the cloud of points is 20microV large (radius) and centered at 40microV.
This means that some residual stray light from the pump is still entering through the OD11 filters in the in-loop PD.

So I moved the in-loop PD to another position and now the noise is reduced from 40microV to 20microV and the cloud center keeps staying at 20microV.

I scanned again the 0.65ppm-absorbing LMA sample. I Plot the AC signal scan and, with dashed lines the AC with phase filter applied (expected phase: -85deg, from the reference scan).
The phase filter removes the component orthogonal to the absorption signal.
The central peak of the dashed orange line is 110microV, which corresponds, with DC=2V, R=7.7W-1, and pump power=10W, Abs = 110/2/7.7/10 = 0.7ppm. In very agreement with LMA measurement (0.65ppm)