The output fiber of the laser is single mode, polarization maintaining.

The reason of the jumps could be related to the polarization or some modes hops. So I tried to rotate the polarization.

Until now the polarization of the beam exiting the fiber was P (I didn't consider this parameter when I assembled it).

I rotated the fiber collimator in its mount by 90deg, in order to have S-polarization.

In this configuration, the loop works better. See the plot

Since rotating the collimator caused the misalignment, I ordered beam splitters and a half wave plate to optimize the polarization without loosing the alignment every time (they should arrive within one week.)

The out-of-loop PD was on the waist of the probe (at the sample position). The final test will be to put the out-of-loop PD on the imaging unit, where it will sense only the central part of the beam.
If the jumps we see in the intensity are related to the modes, and therefore to the shape of the beam, then the loop will not work, because the in-loop PD will see the total power, and doesn't see the shape.