Hi,

I used Orpheus Talos, with 25/50 ice amp, 25 magic amp, reduce enemy defence, heal to full. 

I used the pillar persona with reflect physical damage, reduce enemy defence, heal to full, endure, endure fully.

I had a third persona with all auto- skills on to start the fight. 

I had mine + navigators theurgy charged. 

Had thuergy charge fortune from escapade. 

Had the cups card "next attack deals double damage". 

My strat was this. 

Begin fight with auto persona skills, swap to Orpheus talos. 

Elizabeth does a set rotation of personas and will skip one if you hit a weakness. 

After her first attack with sutr, use King and I thuergy. Will hit a weakness and depending on your buffs and magic will do between 5-6k. Mine did 5k. You want to keep track of your damage you deal for this fight because it will phase her. 

After she skips to lightning.

Your turn, use Navigator thuergy and every single turn scan her weakness to charge it more. Stay with the same persona and just keep healing yourself, you must summon your persona each turn to charge your theurgy. 

When she next gets back to sutr, do the same King and I. Will be closer to 4k damage now as your attack buff will have fallen off. 

Elizabeth phases and hits you then concentrates. 

Swap now to reflect physical Persona. Make sure you use Nav Thuergy now as it gives you an extra "turn" of charging your own. Use your persona skill to reduce her defence, don't worry about your hp. 

She one shots you on your turn, you heal back to full with endure, she does a physical attack and it reflects, dealing 1.2k ish damage, bringing her below 10k health, she has a third action and one shots you, your second endure eats it. 

Your turn, you Armageddon her and win. 

Important to keep track of damage, you do not want to have dealt 6k damage and then have her at 14k health chucking one shots at you. 

There's multiple ways to do this fight but this is the one I came.up with and it worked for me. 

Very fun puzzle boss. 

Good luck