I have some industrial duty double stick tape, I may try that.
As an aside, the MDF doors on the cabinets have some sort of very thin printed faces. Thinner than contact paper. Many of them have slightly puffy faces along the top or bottom edges. I have made a little "rail" with a rabbet in the back. Set up a router to cut a very light rabbet in the door to match, eliminating the puffy part. Going to stick them on with some of the tape and glue in the door rabbet. They will also serve as a pull. Will seal the edge with CA glue.