Couple of ideas have crossed my mind, not sure on the viability of them due to not having the car here to look yet.
First idea is to use either MDF or hardboard to replace the cardboard liner.
Second would be to use extra wide Fablon or similar with a thin foam lining - foam can be found on most markets in various descriptions.
Alternatively go for the 1970's "custom car" look with Dralon and buttons with a thick foam backing - would work well with something like zebra or leopard print but that's probably an idea better suited to a totally different forum!