Early game · first 8 minutes
Use Charged Shot from cover while Rat King crosses open ground. Keep a dash for Rat Swarm; each ground or air dash removes one rat.
Grey Talon vs Rat King
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Mid game · Walkers
Place Spirit Snare between Rat King and your team, then detonate it with Charged Shot as he reaches it. The curse can stop him from firing or activating Royal Pestments.
Late game · Patron
Use Guided Owl to stun Rat King as your team attacks. Save Rain of Arrows for after Royal Pestments breaks or expires so its extra shots are not reflected back at you.