I will try to work with @maik_s offline to try to get something working with the hd44780 library.
But before that, can I get some clarification on which datasheet is for this display?
There are multiple datasheets posted and they are all different for different displays.
If this is the display indicated in OLED application note, then the existing hd44780 library can be used by simple configuring it to extend the instruction times since according to that datasheet the display device is VERY slow for the clear instruction and perhaps slower for the other instructions. (The timing for the other instructions is not indicated)
But instruction times are configurable using the hd44780 library.
If the display is as indicated by the RS0012 datasheet, then there may some additional things that need to be addressed.
So which datasheet is the one for this device?