I think the easiest way to do it will be to rip out all the internals and put a Micro, Nano or Mini Arduino in there. With a screen you buy off-the-shelf you will get the information required to interface to it.
Since I've made a lot of projects with custom circuit boards, I would make my own PCB which holds the buttons. You will probably find that the buttons are relatively simple contacts or maybe you can put a modern tac switch under each one.