Whether or not you cast the copies this way, the original cards will remain in exile.

Copies cast this way are cast during the resolution of the ability. You don't have to follow normal timing requirements and you can't wait and cast them later.

After you pay the cost one or more times, a second ability triggers and you choose the targets for it. Players may respond to the new ability at that point. As that ability resolves, Bloodthirsty Adversary gets its counters and the target cards are exiled.