I know that this is possible on the Gear Mastermorphix, so it should be possible on this too. I've found a tutorial on YouTube, maybe youu want to search for it.
Thanks, misterchris. For some reason I missed Michael Cha's tutorial
in my previous searches (I only saw his demo solve). He covers this case and similar cases starting at 05:35 and uses different (proper) names for the pieces than I used.
After looking at the video, I feel that I was on the right track looking at supercube centre twist algorithms. I will give them another try. I am interested to learn the theory behind the puzzle getting into this state.