The speedrunner achievement requires you to "Complete The Oakshield's Demand under 90 seconds.", which means destroying all 3 cysts as fast as possible.
Since enemy animations take time, it is best to finish the mission in 1 turn.
This guide shows how to do that on hard difficulty setting.
The guide does assume you have some lategame items and skills, but those are good anyway so you probably should have them.

(kills the crypt cyst with a boost from Oiko):
items:

• Warp maul + warp (1 gem)
  
• Dagger + pierce (1 gem)
  
• Ninja robe
  
• Boots of speed +speed (3 gems)

Skills:

• Mobility
  
• Action Surge
  
• Charge + range (1 gem)
  
• Flurry + damage (1 gem)
  
• Shift +4 uses (2 gems)

(you can also take more shifts instead of speed on the boots. Still need the boots for the dash though.)

(kills the second cyst):
items:

• Longbow + pierce (1 gem)
  
• Deer armor
  
• Cloak of night

Skills:

• Mobility
  
• Advance
  
• Power attack +1 use (1 gem)

(boosts Leonhard):
skills:

• Mobility
  
• Teleport

(kills the first cyst):
items:

• Sword + pierce (1 gem)

skills:

• Somersault
  
• Action surge

This is the simplest version of the loadout, some items can be swapped for others.

Move Oiko the farthest you can, then teleport swap with Leonhard.
Leonhard dashes in the direction of the crypt, then moves into the crypt.
Shift onto the pressure plate, swap with nearby skeleton to make it stand on the plate.
Shift toward the cyst, and charge the second rat diagonally, so you can approach the cyst.
Shift to the cyst, action surge and flurry the cyst (6dmg), killing it.

Vanish, move towards the cysts, power attack + advance on the second cyst (3dmg), then power attack+ attack it (3dmg) to finish it.

Jump and move towards the first cyst, attack (3dmg), action surge and attack again (3dmg).

This solution is less straightforward, but has less requirements.
It still requires action surge on leonhard and the Warp maul.
Leonhard:

• Sword + Pierce
  
• Action surge
  
• Charge + range
  
• Mobility / shift

• Warp maul + warp
  
• mobility
  
• advance
  
• power attack +1 use
  
• dash +2 uses
  
• shift +6 uses (3 gems)

• forcebolt + pierce
  
• focus
  
• teleport + range (or mobility)

Mochizuki is not required

Leonhard charges the 1st cyst
Aava dashes and Leonhard moves toward the crypt
Oiko moves forward to get the beastmaster in range of teleport, teleports and moves 1 tile forward.
then focus + forcebolt on the two cysts, breaking the first cyst.
Leonhard moves and charges the second cyst, breaking it.
Aava dashes, then moves into the crypt, then shifts onto the plate, warps with the nearby skeleton, shifts into the left passage, dashes toward the cyst, power attack+hits it, shifts into the alcove and power attack+advance hits it again.

Be careful not to take too much damage from enemy opportunity attacks.
Or add more shifts if you don't want to wait for enemy opportunity attack animations.
Or use the free move / vanish items.