I fixed it, but I'm wondering... "you may be a redneck if..."
The black "snap thing" is as described. It's at the bottom of the seat back between the back and the bench, and it's a relatively simple matter of unhooking it along the whole length at once, no tools required. Then the seat cover can be folded back, like starting to turn it inside out on itself from the bottom. There are some long strips of velcro on each side and under the head rest. These are industrial strength and I had to use a utility knife to free the thread loops from the hooks.
I couldn't think of a clever way to repair the "spring" rods. They were broken at the bottom and almost broken at the top. In the end I cut them out with bolt cutters and ran two lengths of steel wire, maybe 10 or 12 gauge, from the top, wrapped around the bottom, and back to the top on each side. Then I put the seat back together.
"You may be a redneck if..." the driver seat in your pickup truck is held together with wire!
But, it doesn't poke out the back of the seat any more, and it does offer a little more support than it did. Maybe I did good! If I have to do it again I may rivet on some flat stock.