I took 2 pictures of the Menhir spectrum after fiber injection.

Both spectra are centered roughly on 1030nm => it's ok.

The ripples' amplitude in the spectrum is due to the optical fiber being more or less twisted.
It's not really a misalignment of the spectrometer itself, as I tried moving the fiber a little on the spectrometer side.
It seems the effect comes more from the FC/APC-SMA fiber itself...

the periodicity of the oscillations is 2.44nm, which could be explained by a cavity made by 2 optical reflexions (SMA connector / spectrometer optical window?) separated by 200µm.