Bug Fables: The Everlasting Sapling

First and foremost... Block. Poison. Attacks. You won't be poisoned if you hit the timing for the block.

Phase 1...
For this boss, it's a good idea to have Vi as the party leader when you pick up the artifact. As such, it might help to equip her with the HP plus medal from earlier in the dungeon. Tornado Toss is probably your best single-target DPS at this point in the game. For 3 TP Vi can deal 5 damage to the spider while it's up on its web to knock it down for the other two. However don't start burning TP until the spider calls for backup, which it will do once it's BELOW 20 health. Use the first turn to spy on it and get it's health bar and try to end the 3rd or 4th turn with the spider at exactly 20 health so you get an extra turn to pump damage into it before it summons allies and starts doing more attacks each turn.
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Phase 2
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Once it calls backup, have kabbu flip the spiny enemy and follow up with leif to clear the way to the spider. At this point you should focus the spider having Vi knocking it down with Tornado Toss each turn for 5 damage if you hit all the action commands and the other two attacking afterward. No need to worry about the flying mushroom until the spider is out of the way. Using this method you should only be exposed to the second phase for 2 turns, which should make things decently easy.

*Hard Mode can be unlocked at the very beginning of the game by talking to the orchid mantis with the "!" in the top level of the explorer's hall to receive the Hard Mode medal. It has no MP cost.