The information on the internal design of the LEGO kit is currently limited to two images, which don't provide enough information. Contacting the original designer of the LEGO kit may be the best option to gather dimension information.

Determining the feasibility requires the following questions are be answered:

What space is made available inside the LEGO kit when the existing cartridge and lenticular screen is removed?

Is there space under the button mechanism to insert a PCB with membrane conductive pads?

There seems to be a physical separation between the screen area and the button area. Is it possible to run wires or ribbon cables between these two areas?