Ante first. While it is less friendly to the player posting the ante, it is more fair to the rest of the table as a whole. If it's BB first, then then the player posting the BB and no ante essentially gets to skip posting an ante at all. Yet, this person is eligible to win the ante from everyone else on the previous/future hands. Thus, BB first creates a situation where more people get "robbed."
It's def a feel bad for the person posting to only be able to win their ante back. But imagine a situation in a normal ante game where a person loses a hand in the SB and is left with about a single round of antes. Once it gets to the point where he is all in for just a single ante, the amount he can win is a round of antes. Which is the same in the BB Ante format posting Ante first; he wins back his ante, which is a round of antes.
Don't forget that this is somewhat balanced by the fact that if they win their ante back, they now have a full round where they can win everyone else's antes+ their current stack if they get all in.