Turned out this wasn't as hard as I expected. There are brackets on each side in the back of the sofa that hold the part of the frame that moves up and down. Those brackets have anchors/inserts in the side for 1/4-20 bolts. On one side the bolts were gone, and the bracket and the sofa back were just drifting around (which is why it wouldn't lock closed). On the other side, the bolts were loose.
To remove the back it was necessary to pull the cotter pins and remove the clevis pins. Then I could remount the one bracket and resecure the other. On the side that hadn't come loose, there were no lock washers-- so I expect there weren't on the other side either.
I replaced with new stainless 1/4-20 bolts, adding lock washers AND blue Loctite. Hopefully the sofa will stay in one piece for a while. After the repair, there was no problem in having the sofa lock into place. A road test will give the final verdict.
I didn't take a picture before taking out the back. These show the brackets and clevis pin.
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