No, it will not trigger. Since the spell is now an Aura and not a creature, it does not have any creature subtypes.
Nameless Inversion is different because it has the Tribal type which allows a noncreature spell to have Creature subtypes. In the case of Inversion, Celestial Kirin will trigger.
If I cast Eidolon of Countless Battle as bestow with Kirin in play, does the Eidolon still count as a spirit spell? I presume the ruling would be similar to Nameless Inversion?
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No, it will not trigger. Since the spell is now an Aura and not a creature, it does not have any creature subtypes.
Nameless Inversion is different because it has the Tribal type which allows a noncreature spell to have Creature subtypes. In the case of Inversion, Celestial Kirin will trigger.