To answer my own question, yes, a lot of stuff had to be moved. I was about to abandon the project, but my intrepid mechanically-minded daughter forged ahead to success.
First we removed the overhead A/C vent panel, held in place with four rear-facing phillips screws along the back edge, and spring clips hidden overhead. We removed the screws, then popped the panel down.
We thought that now we could get to the upper phillips screw along the edge of the hatch, behind the A/C up-duct, but no, the A/C up-duct was held in place by the A/C panel under it, and that was held in place by the panel in front of it. So to remove one, we had to take out all three.
The forward panel is held by two hex-head bolts in the outside wall of its storage compartment. We removed those with a ratchet wrench, and that panel came loose. The rear panel is held by one allen-head bolt in the outside wall of its storage compartment. We removed that with an allen wrench, and that panel came loose (after popping loose the spring clips along the back edge), exposing two hex-head screws holding the bottom of the A/C up-duct panel. We removed those two screws, the A/C upduct came loose, and now the long-sought-after phillips screw on the hatch's edge was exposed. I realized that with the correct screwdriver (L-shaped or even S-shaped), we theoretically might have loosened that screw, but it was extremely tight, and required a lot of weight and torque to break it loose with a regular phillips screwdriver. (When we replaced it, we left it just a little loose, just in case . . . )
Once that screw was removed, the entire inside molding on the hatch could be popped off (it took a little effort -- the spring clips were stiff). The upper screw on the other side of the hatch was still tight, still obscured by the panel in front of it. The final effort involved wedging the edge of the panel by the door gasket, gently using a flat-headed screwdriver.
With the molding pulled out a few inches, I could get my fingers inside a little window at the corner of the hatch, in front of the malfunctioning lever assembly. After a few minutes' fumbling, I was able to slip the L-shaped end of the rod (extending from behind the hatch handle in the middle of the door) into the socket at the end of the lever that trips the door open. It had apparently simply fallen out. Now the door worked fine.
However, I was concerned -- the end of the rod fits through a nylon split-grommet in the hole in the end of the lever. It seemed to me that the rod was too slack in the hole, which would probably explain why it fell out. The rod/grommet assembly on the opposite corner of the hatch seemed considerably more snug. The end of the rod had a groove around it, so I decided to make an additional washer contraption to grip the end of the rod at the groove. The only thing at hand was a plastic boxtop off an oats box. I drilled two holes in it, trimmed it out as two strips with a hole in the end of each, and popped both of them over the end of the rod. (I would have used just one, but it still seemed to have a little slack after I mounted the first strip.) Now the assembly seemed snug, and still worked fine.
Then we reassembled everything, in reverse order. I used crazy glue to repair the broken end cap on the hatch's inside handle. It seemed to pop back into place and hold OK.