Jon,
Not sure what the customer is expecting for the end results, but from past experience of work in restaurants, here is my input.
We wanted the grease and food removed, so they were not slippery through out the shift. They get pretty slick anyway and after not being washed well, they get pretty bad.
We used to take them out back, drape them overe the wall and hose them off with hot water run from the water heater, brush the stubborn areas and let them dry. We did not use soap.
If you can't hang them, I would wash down the area that you would lay them in first, some you can stand on their side and lean on a wall, some you can't.
With the heat and pressure of your unit, it should be a snap to do, but moving the dirty mats is a chore, so wear old clothes and or raingear to keep clean and dry.