I’m not sure that I understand your question. You can share metadata between Bridge and Lightroom, but it starts by saving metadata externally from Lightroom. Bridge is only a browser or viewer for the contents of your computer. So if there is some metadata in a folder, or embedded in a file, Bridge will see it. Lightroom is a database, so it saves the metadata inside of the database. You can save Lightroom metadata externally by turning on the preference to save changes automatically, or you can save it one file at a time with a CMD or CNTRL S.
Here’s a video I recorded about metadata conflict warnings that might help How to deal with Lightroom Metadata Conflict Warnings - YouTube