Your idea solar GameBoy looks great too.
It seems Embox fit for your purposes as an operating system which allows running a lot of ready to use Linux software and at the same time save hardware resources.
Embox in the one side can run complicated Linux applications Qt (https://www.youtube.com/watch?v=9rA_sHrQjjE), PJSIP (https://youtu.be/fJ-HNopjQkM), or OpenCV (https://email@example.com/how-to-run-opencv-on-stm32-mcu-b581f42b0766) and the other side doesn't use excessive resources. EFM32 for example. Except this, Embox run on Dialog board with BLE, USB, HTTPD, TELNETD, and have < 50 uA current consumption of course if you are not using the applications