How can I move files or directories marked as a backup?
If you list a file inside the TM backup with the Terminal command ls -l@, you'll see that it has extended attributes:

$ ls -l@ "iTunes Library Genius.itdb"
-rw-r--r--@ 22 admin  501  94208 Sep 17  2013 iTunes Library Genius.itdb
    com.apple.metadata:_kTimeMachineNewestSnapshot     50 
    com.apple.metadata:_kTimeMachineOldestSnapshot     50 


I guess you'll have to remove those from the files that you moved outside the TM backup folder. I haven't tried that myself, though, as I do not want to mess with my backups :)

To remove these attributes, use the attr command with the "c" option, like this:

xattr -c /path/to/file


You may have to use the bypass command for that again.

Let us know whether that works.