After checking the alignment of both the probes, I tried to measure a LMA coating (that absorbs a few ppm).
I increased the pump power up to 1W (980mW) rotating the HWP in the IPC (so without changing the laser current the power is immediately stable).

The looking at the scan on the screenshot attached we can see that there is a large constant-phase signal.

After removing the sample it was clear that it is stray light from the pump because the phase is -22deg.

In front of the PD there is a long-pass filter 1250nm that has OD 5.5 @1064, but it is not enough.  The transmission at 1310nm is 85%.
Probably the fastest solution could be to put 2 filters together in the same SM1 attached at the PD, but I'm afraid of internal reflections effects.