I'm actually not sure if we're the ones who kill Brynhildr, since her part hasn't been officially translated yet. It's actually coming out in a few hours since the event is going on right now, so I'm looking forward to it.
In any case, looking at the other fake characters that we battle, they all fade away after we defeat them. However, it's stated that they all had a short lifespan, being fake, and none of them really have a problem with dying. I'm assuming it's the same with the fake Brynhildr.