I assume that the ribbon connects the processor to the LCD display. It's probably not going to help you read the buttons directly.
Reading the buttons might be easy. If you're willing to build a new PCB using a cheap service like OSHPark, then you could make a compatible board with the same screw holes and align the sensors under the buttons. You would probably only need a 2-layer board (very cheap and quick) and you could add a connector to feed the Axoloti.
Writing to the display is probably going to be much harder. So long as the original processor board is attached to the display, it will be difficult to change what is displayed. If you disconnect the LCD, then you'll have to find out which pins are power and which control the content. If there is a part number on the display, then perhaps you can find documentation. If not, then you have a lot of hacking to do, and it might result in damage to the display if you're not careful. Also, disconnecting the display could damage it. It looks like this thing is soldered together and not meant to be serviced or replaced.