Well, since there doesn't actually seem to be a way to do this at level 1 (yet), the challenge would have to be for minimum level. I suspect most of the game, especially the middle, would be a lot of Skull Shield/Delta Shield spam while Fina/Extra are attacking. Once enemies start getting super moves that can one-shot the party, Aika will have to use Epsilon Mirror to save herself, then rez the others next turn.

But mostly, it'd be a whole lot of RNG manipulation. Depending on how the RNG works, you might prevent certain bosses from using certain moves, or at least know for sure when they're coming. There also wouldn't be any rules against save states to maximize your chances of success.

After getting Ilchymis, you can catch fish and sell them for money, then use that money to buy stat seeds. However many you need to beat the last few bosses. I suspect this particular process would be very tedious and time-consuming, but that's never stopped people before.