I saw a vid years ago from a woodturner who turns very small stacked laminated pieces (1/16th - 1/8) that was separated by small wood bars. He placed the glued up blank into a coffee can and poured in heated/melted paraffin wax. The wax filled in all the small spaces and provided support to the very thin wood. Basically the flimsy blank became a solid blank. Once cooled he would mount the piece on the lathe and turn his design. Once done, he would use a heat gun to melt the wax from the fragile shaped turning.
I know this is not much of a description of the process and I am not sure how well it would be used in your bowl project. I could see building up the segmented piece to a logical stopping point, doing the wax thing and turn to shape. Then cleaning up the top, adding layers, and repeating. Just a thought on how to support very thin pieces for turning.
I will admit I have not done this process yet.