I have a similar issue.
How are you copying the files? The simplest answer would to see if that method supports keeping the file modification attributes. MusicBee will recopy anything with a newer timestamp, so it might be seeing a different time stamp than what's on the HDD, think it's changed, and then recopy it.
I know FTP and WebDAV servers hit this limitation. There are FTP clients that can do it, but not the build in Windows support.
If you are plugged in, then MTP should match those attributes correctly, in which case we are probably both hitting the same bug.