Well, not sure how BST would play as a limited job. BLU learns spells and breaks a few older content. BSTs would be severely limited to overworld enemies. They'd have to introduce a Pokemon style system where they can fight, capture, and train their pets in an arena. As the pets grow stronger they can do harder content to capture even stronger enemies from dungeons/raids.

Luckily they do have a system in place they could take from, and that is the chocobo companion. While the chocobo is far more grindy to get to max, pets shouldn't need to take that long.

Having BST a limited job would stink, but it would breathe some life back to BLU for a while since BLU content has a FOMO if you don't get in on it while it's fresh.