Cat would not have to die again for her to bring Brienne back. Beric was brought back numerous times from death and what killed him in the end was bringing Cat back from the dead days after she had been murdered coupled with him being resurrected numerous times. Remember when the Hound killed Beric in his trial underground? Thoros takes him away immediately and gives him the Kiss to bring him back. Cat was dead for days in the river when she was brought back. I think it becomes harder to reanimate the dead the longer they have been expired and because of this evidence Cat would not have to die to bring Brienne back from the brink.

As to how she would have the power I would have to venture a guess. Beric Dondarrion wasn't a follower of the red god like Thoros was, they were just friends right? I could be completely wrong there I admit, but wouldn't it be mentioned before if a powerful Lord wasn't a worshiper of the new or old gods? And according to Thoros, Beric was the one who Kissed her to life and then died for good. I don't see why Cat couldn't do it now as well without losing her life. Shit isn't Thoros still with the band without brothers or whatever the fuck there called? He could have done it himself.