I enjoy making these cabinets and have made 4 variations for my home (one has double doors).
On making my second I had the door made and the shelf depth. I hadn’t made or accounted for the additional depth to the door of the wooden battens fixing the mirror. I didn’t want to reduce the already narrow shelves.
To make the door profile/depth slimmer I used 4 picture hooks. I remove the angles where the nails go and leave the hook. This leaves you with a J shaped piece. Turn it over and enlarge the remaining nail hole for a small screw. The hook then acts as a spring against the mirror/cardboard/ply. I fix them on the sides an inch or so from the top and bottom stiles.
The double holed larger hooks work well, although no issues with the singles.