If you don’t get a solution here I would recommend asking on bazzites discord or forum. There are more helpful people with more insights into immutable there. If it can‘t be solved without layering they might even be open to having it included in the image by default.
On iOS you can enable Guided Access and restrict what one can do, for example disable touch and lock it to an app, until you enter a Code. I imagine Android will have something similar.
This obviously doesn’t protect against electronic forensics, but it does protect against just opening different apps and searching through the phone manually.