I did a bit of cheating in Dorico, and it worked!
I needed to have a snare staff and a woodblock staff, and cross-staff notes between the two.
If I simply moved the notes it worked, but the sound was wrong (obviously).
So what I ended up doing is creating the extra notes (the ones that would be cross-staff) in the snare drum staff, then I suppressed playback for those notes.
I then hid the stems for the woodblock notes where the cross staff would appear. The noteheads remained, just the stems were hidden.
Then I proceeded to simply press āNā to move the required notes.
Now I have both proper playback of both a snare drum AND a woodblock, while at the same time having the correct notation for this passage.
I have to say I am VERY happy with the result (though the staves are a BIT too far apart for my taste. I will probably be fidgeting with those a bit to get the correct amount of white space.