I'm late to the party on this, but I would agree with the others that Raspberry Pi is not only overkill, but will make this more difficult than it needs to be.
It's a great job for a basic microcontroller, and the code needed for that will be simpler. You just need something like an Arduino, some wire, a few resistors and 4 buttons. Look at any intro to Arduino introduction that gets to button presses (and debouncing). Here's a good guide from the start: https://learn.adafruit.com/ladyadas-learn-arduino-lesson-number-1/introduction
If you want to follow a guide end to end and can customize the code, this might be a helpful starting point with a very compact board: https://learn.adafruit.com/arcade-button-control-box/overview
Looks custom. It seems unlikely that Lenovo would offer this.
Run your fingernail across the transition in each direction. Is the carbon fiber part higher than the rubber coating, or lower?