You might be able to scrape off the soldermask (the green layer) and solder directly onto the connector. Though, that's probably destined to fail. What you might want to try instead is to transplant a Gameboy Pocket display onto the display board. This would still involve fine soldering, but at least you'd be soldering on surfaces meant to be connectors.

Hmmmm, a Gameboy Pocket LCD eh???  I just took a quick look on Google & already found some interesting information on that.  Thanks for pointing me in that direction bro.  I'm all for hardware hacking, & if I can find a way to re-engineer this screen setup, I'll gladly do it & document it.  I'll start researching once I get a little more time.  I'm keeping this unit for experimentation.  Hopefully, something good will come of it.

Seeing how impossible that ribbon cable is to work with, I attempted to solder some wires from the pads to the ribbon cable.  However, soldering onto a ribbon cable proved too frustrating for me, even after scratching off the coating to expose the traces.  I gave up.  So I made sure to take some detailed, close up pictures of the motherboard, traces, & everything that's behind the screen.

I hope these can be useful to somebody who wants to plan & experiment without destroying theirs first, or something.  Or even just to document.  Feel free to take these pics & use them as you need.  You can get bigger versions on my photobucket page.  Each picture shows the link.  It would be nice if someone could come up with a different way to connect this little screen... I hate the way they set it up.

FRONT MOTHERBOARD

This is what a good one looks like.  If you're backlighting your DMG-01, please do be careful not to bend this ribbon cable too much, or you will suffer like me. 

REAR MOTHERBOARD