Really how I'd expect Beastmaster in FFXIV is as a melee version summoner where some attacks would cause the pet to strike with you and have some pattern of BST attacks buffing the pet and pet attacks having a chance to increase the BST's damage or something along those lines, and it would have at least one cooldown to spawn extra animal companions to fight for some limited duration.

Puppeteer would likely fit that mindset of "QQ I want a real pet class", where it would look like it's shadowboxing while the puppet does all the actual damage. On the plus side, it would make for a more interesting alternative to physical ranged damage-dealers.