Not sure it's that different.
Some people use a scrap blank and rough turn a shape to approximately fit inside the bowl to be reversed. Then cover it in Router Mat, kitchen paper towel etc., to take up the slack.
I came across the Kingspan/Celtex insulation one day and acquired a piece thinking it may have a use, even though I could net see what at the time.
Since then I've found several uses for it, but this one seemed to answer you request for an idea on reversing a bowl.
Other uses are Goblet reversing, pincushion filling, small items can be pressed into it etc..
As a separate thought, if your bowls are 12/13" diameter, the Doughnut chuck would need to be 16" or more in diameter to accommodate it. And it would not be able to hold Natural Edged bowls easily, if at all.