I would give companions with Initiative 1 their own turn controlled by the player. Brave Beast has Initiative 0, but it can attack with the Beastmaster thanks to the special ability as you noted, so I think it’s powerful enough.
I think if you or someone in your group feels that this is too restrictive or no fun, it is more important to cater to that and talk about what seems fair, rather than doing what I would do at my table. If you are running the game, maybe you can arrange for them to go on a quest or find a magic item that grants their companion 1 initiative.