I then click Refresh and beatunes does not clear the correction.
I'm not sure I understand this. Which "Refresh" do you click? And why should beaTunes clear a correction? Wouldn't you want beaTunes to pick off the changes in Music? For that you'd have to synchronize beaTunes (in the File menu).
Is there any other way to get these changes to be seen without having to perform another synchronize?
Synchronization is the function that does pick up such changes. Why would you want to avoid it? Too slow? If so, how large is your library (in song count, not GB or MB).