However, watching some videos on Bitwig's modulators ( in particular) I came up with this simple idea to work around it. It's far from perfect: works only in range of +/- 1 octave and the quality depends on the sounds used, but it's better than not having it at all.
Basically:
- set the Sampler to not follow the note keys (the piano icon in top left),
- add a Key Tracking modulator to the Sampler, set it up as Relative with Spread of 12,
- make sure that both Sampler and Key Tracking have the same note as a base/root, e.g. C3,
- put the Pitch Shifter in Sampler's FX and modulate its Pitch knob by the output of the Key Tracking,
- adjust the Grain on Pitch Shifter to your liking, although I find the default value of 10kHz (I think?) to sound the best.
I hope the above is pretty much self-explanatory, but if not I'll post a screenshot in the evening.
This way the Sampler will now play your sound +/- 1 octave around the root note in response to your keyboard, changing the pitch accordingly but at a constant speed

