I think a folder would work fine as long as you had wide enough tires, something like 35mm or wider to give a softer ride. I was pleasantly surprised a year ago when I rode the GAP and C&O at how well maintained the trails were, but I used a bike with 26X2.0 tires. I recall a couple short sections of maybe 100 yards near communities where the trail was paved and the paved portions were worse than the rest of the trail.
I am not familiar with the Greenzone folder. If you plan to camp, would this bike carry all of your camping gear well?