I wonder if it would be acceptable to mount the receptacle to a working cabinet door, and connect it with flexible armored cable?
If the ahj allows it but I doubt it. If you can demostrate the installation and the components that are used are listed or approved for the application, you can demostrate there is no other reasonable solution, and if you can demostrate that the installation is safe installation then you may have a chance. However, being is a swinging door, cords are subject to damaged or getting yanked on.
Years ago I was helping on a high end house and the Mrs. wanted an outlet in a drawer for her hot rollers. The AHJ allowed it. We cut in a box in the back of the drawer and attached with heavy use SO cord and good quality cord grips. The side rails for the other draws provide a protective cavitity for the slack of the SO cord to ride in. It was a beautiful thing. Although I would not normally recommend such an installation, everyone did their part and pending on the situation, I would allow it today providing it did not pose any un-reasonable risk. BTW, the hot roller holder was warm to the touch on high and it had a built in over-temp protection and the outlet was a GFCI.