limited job
weapon - axe and shield
skills - change based on beasts out
concept - a job that levels beast and uses them to fight alongside them
interactive/rewards - beast tamed and leveled to a certain degree can be used in the companion system. Maybe even a bonding system requiring other factors to allow using them outside of the BST class.
Idea being you level and train your beast so you can use them on your main with different beast, unlike the chocobo we have, only having a pre-set skills so you would ideally want to tame different beast for different situations. ie behemoth being a tank monster only able to tank, or a couerl being a damage dealer, etc.
Some colosseum stuff with both pve and pvp content where only the beast can fight and you command them resulting in glam rewards and such for accomplishing goals. not to different than BLU.

I think the companion system could really use some work to expand and the BST job would be a great opportunity to do so.