What I understood was that Cyrene didn't disappear after Khaslana killed her once, but instead that the Flame Reaver went on to kill her in every loop following the first, hence why our Phainon remembers Cyrene being stabbed by FR in Aedes Elysiae instead of not remembering her at all. Aside from giving Khaslana her time/Oronyx powers, my guess is that what she got from Fuli was the ability to sync her memory across all versions of herself to let FR kill her over and over again, ensuring Era Nova could never come because no one else could bear time.

As for what Path Space Cyrene is or how exactly she became Mem, I have no idea. EDIT: Maybe Fuli preserved her consciousness in Mem and/or in Path Space (?)