Managed to beat Medusa Ragnarok without an Elixir. It really was just luck on the Petrifies though. First Petrify missed the Soul, so I was able to cleanse it with Maiden's Prayer. Second Petrify I cleansed with Behemoth. Didn't get a third Petrify before my Burst on turn 5. Stun gave me 3 turns with no chance for Petrify. Got 1 turn without Petrify in after that where I used all my skills, then finished her with a Partial Burst on turn 9 for the kill.
The damage stacked up from her Petrifies and Rage Overdrives, and that's what ended up killing my Soul. Might run with Jorm as primary eidolon next time to provide a bit more lasting power. In any case, I'm fairly sure I would've been doomed if even one of those Petrifies stuck. Didn't help that I was running Maiden's Prayer instead of Sniper Shot, giving up 10% ATK and DEF down.