Naturally, I was fighting with this for hours, and answered my own question a few minutes after sending my query. It looks like I need to copy the relevant heldmsg-$LISTNAME-* files from /var/lib/mailman/data along with the request.pck database. Apparently, when the CGI script queries the request database, it gets and displays all the entries there, but also notices which entries don't have corresponding heldmsg files and deletes them.
Oh, well, maybe this question being in the list archives will mean the next person will have more success than I did googling to troubleshoot.
With those files copied as well, everything's looking OK.
Jay
-- Jay Sekora Linux system administrator and postmaster, The Infrastructure Group MIT Computer Science and Artificial Intelligence Laboratory