OK, so it looks like I dodged a bullet on this one (for which I am very glad!).
I took the approach of "If you're not sure that doing something could break what's currently running, the don't do anything." I let the NeoLoad test and monitoring continue to run (it was a 5.5 hour test and most of it was concerned with gathering performance stats). The "Saving Project" modal dialog stayed open the entire time (about 3 hours from when it first appeared). Since it was a test with a set duration, NeoLoad gathered the statistics and saved the test result. So that was a relief.

But I still didn't have a saved project after that was done. After the test was over, I tried to restore an old (valid, from a very recent project save) config.zip in the project's directory. As I expected, this didn't really fix the error in the running GUI. I had to then use Task Manager to end the NeoLoad GUI process.

When I restarted NeoLoad, my results were all there from the 5.5 hour test. Then I was able to save the project and I only lost a few changes to the project.

Additional info: In the neoload.log file, I found the original error that had popped up in the GUI: "ERROR - neoload.Configuration: Corrupted config file java.util.zip.ZipException: ZIP_Read: error reading zip file."