Sophia's Story mode works differently from most others. It consists of 12 Stages, but each stage is played individually and has a task assigned to it. Winning is only 1 of 3 criteria for gaining completion. The other 66% is made up of completing the task and an extended version of the task. For example for one stage you must win the battle with >60% health, the extended task is >80% health. If you complete it or not you are always taken back to the Story selection screen.
33% - Win
66% - Complete the first goal
100% - Complete the extended goal.
The good news is that you do NOT need 100% completion on all 12 stages to unlock the trophy. But you DO need to unlock all the stages by increasing completion rate. The minimum needed to play all 12 stages is to have won on every stage (33%) and then 5 additional stages must be at 100%. Pick any tasks you feel more comfortable with. The trophy will unlock once you have won the battle on Stage 12.