I like the idea, but feel that it can be expanded on. I would much rather it be a job on its own than a feature that any job can imitate. That can be either limited or standard. Though when I think of BST, I think of actively going out, taming and observing beasts to know their traits and abilities. Like a monster journal where after a time battling against or with a species you learn more about them. Maybe even a psuedo reputation system where you improve your relationship or knowledge with a certain species.

That also means every species should have differences besides on how they look. For example: crabs are a sturdy bunch and would make great tanking pets. You start of with a lowly crab that you can find in La Noscea or Thanalan and as your relationship improves it gets stronger and learns new abilities. Then you have more supportive beasts like the mandragora who excel in debilitating enemies with poisons and giving allies regen effects. I like the idea that as they get stronger they can "evolve" into stronger versions of their species. I would love to see a crab turning into the B rank Helmet Crab or the mandragora into a Royal with their own noticeable differences between the "lower" and "higher" members of their species.

It's beginning to sound a lot like World of Final Fantasy. Which kind of pushes it more into the limited job camp.

Of course this will require some fine-tuning of the AI or maybe organize special abilities as commands that only the master can give. Would suck to have to micromanage the cool actions like you have to with the faeries and primals.