When is she out on a failed tag up

[Dakota]

She would be out as soon as you recognize the appeal. If you know the tag is an appeal of leaving early, call the out then.

Yes, they could have requested TIME, and when granted, made the dead ball appeal. Since the runner had stopped on 3B, she would not be allowed to return during the dead ball. For a dead ball appeal, they would not need to do anything in particular with the ball before or after requesting TIME. The player making the appeal does not need to have possession of the ball.