The reason players are taken to another location for the re-challenge is because of issues with the "battleback" (the scenery of encounters). If unchanged, the battleback would reflect whatever location the player was at when they used the Medal, which would be a problem since non-encounter maps don't have battlebacks. Instead, the player is taken to an "arena" of sorts and is automatically returned to their previous location once the fight is over.
To prevent farming (for money or Exp) the re-challenge versions of bosses give no rewards upon defeat. To prevent the player from using the Medals as a way to refill HP and MP, neither of these are changed from what they were before the player is transported. If Ami (for example) has 600 HP and 200 MP when she uses a Medal, and 400 HP and 120 MP when the re-challenge is over, she'll be brought back to her initial location with 400 HP and 120 MP.
If the player should lose the re-challenge, they will not receive a Game Over, but will instead be returned to their original location with every member who "died" having 1 HP.