I haven't exactly finished it all yet. But my end goal is the same as yours. What I originally planned to do, and still haven't given up on. Is modifying the hinges. When I reinstall a few things here next weekend I"ll try to snap a few pics to help explain. But here it goes...
Before cutting space into the box for the screen, I cut out the parts on either side that the door mounts to. That is, I cut a large square out of each side so that all of the mounting holes / clips / etc were still intact with each other. Once the screen was installed and fastened in place where I wanted it I used plexi-glass to re-mount the squares with the mounting assemblies to the box so that they sat on either side of the screen. Basically, the plex-glass is just a spacer between the mounting squares and the original box sides. This way I didn't have to modify the screen casing at all.
My next move was to cut the door hinges just above the circular parts. I then used more plexi-glass pieces as spacers to move the round parts out and make them wide enough to fit on the widened box.
Once I got them all set, the door re-attached to the box and everything worked swell.

However, I quickly found that the modified box/door combo was too wide to fit back into its original spot. Or rather, I could get it to fit, but once in I'd never be able to get it back out again.

That may be acceptable for some, but I would like to be able to get this car back to stock if need be.
So I'm trying to figure out how to take the gray panel shown in the picture above off. That will allow me to slide the box/door combo in and hopefully figure out a way to be able to remove them as well.
If you come up with any ideas of your own I'm always looking for someone smarter than me to give me answers.