Thanks inglisd! I was able to SSH into my BDP-3 and poke around.
The directory where the music on my internally-mounted SSD is /media/Music. Fortunately, this directory is readable and writable by all, so we are good to move/add/copy files in and out.
If you insert a flash drive on one of the external ports it will show up as "/media/Samsung-USB" or something similar based on the flash drive manufacturer. I ran a USB3 extension cable from one of the rear USB3 ports to get easy access to USB3. The two ports on the front of the BDP-3 are only USB2 and too slow to transfer files.
If a user has some experience with the linux command line, moving files on/off the flash drive to the internal SSD is easy.
I backed up all the files on the internal SSD to another drive via rsync in case the internal drive goes bad or the BDP-3 stops working. I'll post more on that later.
Remember to unmount the flash drive via the MM system interface. We can't mount/unmount anything without root account access, and we don't know that (yet).