Yeah, there is lots to this...I hope I am not coming across as condescending because, while I have written a couple of simple display libraries and have used quite a few LCDs with PICS and Arduinos and a few others, I am by no means an expert.
To me, ya got your LCD, and you got your controller chip (KS0713) and you got your driver (in firmware in your case). Both the E and COG "boards" have the same controller (on glass in the case of the COG variety and on the board of the E variety). Somewhere, you have an MCU that sets up the controller and then interacts with it regularly to display what need to be displayed. I am going through this to be on the same page...well, more accurately, because that is how I am viewing the situation. IOW I think about the "driver" as the code that interacts with the controller.
The KS0713 datasheet is here
https://www.crystalfontz.com/controllers/Samsung/KS0713/
You mentioned that you hooked up a different board and it worked but was some kind of mirror image...that is actually a good sign in a way. Same with a recommended display that was compatible except for something like that. I remember an ST controller on a board that had a hardware signal to invert the screen or something similar. So, my point is that, yes, if it worked except for that anomoly, you quite possible could fix it in software. But, how do you get at it inside the product because that is likely a major obstacle.
So, my thinking is that you need to replace the LCD + controller and you need the same interface connections. It becomes a nightmare when you don't have the schematic to your product. The COG display *could* work, but there are probably a few reasons why it might not. Again, if you had a schematic that showed the interface to the controller, it would help.
Sorry, I know that this is not all that helpful.