Just a quick follow-up in case someone comes across a similar issue in the future:

After further testing, I found that CGI::Simple actually didn't quite work for file uploads without modification. I initially did not enable uploads (turned off by default in CGI::Simple), so the upload function was not being called. New files were being created in the filesystem, but remained empty.

CGI::Simple uses IO::File for file handles, which requires dynamic loading. I was able to remedy this by modifying the _save_tmpfile function to not require IO::File and instead use open() to create a temp file. File uploading now seems to work fine.