I just spoke to an Apple Support rep and they told me that this is a know limitation with Catalina. That sending files from an external drive is "no longer supported" (!) and that the file should be imported in other ways, such as from Photos, etc!
Frankly, I don't buy this explanation. This just smells like a BUG. I can't imagine what difference it makes that the file is located on a different path. As long as the disk/volume is mounted and the permissions are correct, there should be no technical difference in accessing the file to be sent. And if Messages can't access the file it should just report it immediately (before attempting to send it).
Even if there is, indeed, a good technical reason why files on external drives shouldn't be sent through messages, then the right thing to do is disallow any actions that attempt it. For instance, dropping the file onto the Messages window should fail immediately with a message telling the user that the image can't be sent from that external drive. Attempting to use the share menus should do the same.
Not telling the user beforehand and just letting the action fail, inexplicably, after a few minutes is very frustrating for the user and just not "The Apple Way". Besides, there's no way to figure out why it failed, even after the fact!
Here's hoping that the Messages team pays attention to this. The Apple Support Rep recommended opening an issue at https://www.apple.com/feedback/messages.html. If they get enough people reporting this maybe they'll have to acknowledge it or, better still, fix it. So if you have two spare minutes, just open that link and file a report with your feedback.