I’ve got almost the same setup as you (RC-300, VL3 and Beatbuddy… and a Line 6 Helix.). I just got my BB (I’ve had the RC-300 since Christmas), and when I hooked them up through the MIDI splitter cable, the BB starts and stops with the RC-300, changes tempo and syncs perfectly.
I’ll make a list of all the settings on both units if you think it will help. I did make sure the firmware was the latest available in both boxes.