I don't have this issue, the chosen picture stays between track changes. In the choosing a picture optons, I use "link to the original picture file"--is that how yours is set? Is the picture on a local drive and a path you have full access to? Are you using the latest 3.4x version of MusicBee?
If the answer to all my questions is "yes," please post the URL of a stream you are noticing this on, unless it's ALL streams.. in which case I can't help you.