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Using a card seems to guarantee encountering the outlaw, but I only used two cards. Also, the bounty hunter always takes the same appearance regardless of card level, using an arbalest, and dealing damage in the 700s. He also has a significant amount of health, comparable to an accomplished player.

He likes to spend a lot of time walking around, and he will actually target anything in the area, not just the bandits. Somewhat annoyingly, he will also choose to target something far away once his target is killed, only switching targets when attacked. Also he won't attack outlaws themselves (most likely just a bug).

I highly suspect that, similar to a shadow crystal or commerce letter, the "BH" effect stays in place until the goal is accomplished.

Here I am listing the BH symbol associated with the cards: