You could divide the screen into rows 12 blocks wide, each powered by an array, with a 2 block gap. You put the arrays one level below the display level and push the blocks up (via sticky pistons) each turn. You'd still have to manually fill in the gaps, but that's only 22 out of 160 lines.
You can combine the arrays like a big stair and only have a 1 block gap, but that requires some manual working of the pistons each turn (because you can't hide the wiring). Not sure if it's worth it. I'm 30% confident it can be automated without exposed wiring.
I have been thinking about a gapless way on-and-off over the last 2 days. I don't have one yet, but I'm 70% confident I can figure one out without the help of the r/redstone hivemind in less than 50 hours of thinking. I've put building a working implementation on my Minecraft todo list. There's no way this is impossible.
... and I've build a working prototype. Took about 3 hours to figure it out, 2 hours to get the wiring to work (first big redstone project), another 1-2 hours for the array and timing. It's trivial to scale and can be easily extended to push in both directions. The whole mechanism is hidden. I think there is a delay of ~8 seconds per 12 blocks, so scrolling the gameboy screen should take ~1.5 minutes. I'm sure you can get this below 1 minute if you try.
Here's the video. Here's the save. Here's a bunch of screenshots instead of a blueprint or explanation.
Here's the new thread for posting quotes, with the usual rules: