TDA 2019 RP-11: Ante Formats. No Ante Reduction
For single-payer systems, the big blind ante format (BBA) with ante-first calculation is recommended. Antes should not be reduced (including at the final table) as play progresses in the event.
Following the above: Yes, the BB should pay the ante first. The remainder, if any, is the big blind amount. In other words, if we change your scenario such that the BB had 500 left, then 400 would go to the ante and 100 would go towards the big blind. Sticking with your scenario, the ante is 400 and the big blind is zero.