I can't see any dye penetrating far enough to be still there after hollowing out. Even if you build it up in rings you will be taking out 10mm or more in places. People do odd things with coloured pencils and epoxy but I never have, search for pencil bowl for ideas.
I wonder if you could make a bowl conventionally with thick sides, sanded but unfinished, cut it up on a Bandsaw, stain each piece, reassemble and just fine sand/polish back on the lathe? Have to think carefully about the blade thickness but it might all work.
Or temporarily glue up your boards with paper in each joint. Turn, pull it apart, stain and rebuild.