Gear swapping is retroactive damage changes on DOTs but not Owl. Owl creates a new entity that seems to copy your stats at the time, so changing your gear will not affect the Owl damage (this is good, they’ll continue to do high damage while you switch to a blunt weapon). Swapping does retroactively affect things like Incineration and Cure however.
For general world bossing, it’s (rod):
Divine Might
Carve Owl x2
Cure
Incineration
Chortasmata
Carnivory
Heal x2
Hotbar swap to (blunt)
Carve Attack x3
Hotbar swap to (rod)
Divine Might
Auto until cooldowns fall off.
Skills off cooldown (make sure Carnivory is up or about to be up if you do decide to use Chortasmata).