So here was the work-around:
I created a new project file and transferred all of the P2 footage using FCP. Since FCP was looking for the QT reference files (named, for example, “p2clip1.mxf”), I then set up a script to add .mxf to all of the file names.
When I re-opened the project file, they all re-connect fine. There was some issue with a couple missing files, but these all seemed to be incomplete clips for which we had the whole clip for. (An issue with card-spanning clips the way they were initially brought into FCP, I think.)
There was probably a more efficient way to do this, but we have it all.