The relevant part of awakened spellbook is:
When you cast a wizard spell with a spell slot, you can temporarily replace its damage type with a type that appears in another spell in your spellbook, which magically alters the spell's formula for this casting only. The latter spell must be of the same level as the spell slot you expend.
My reading of the description of animate objects is that there's only one damage type that's part of the spell, and the others are just a suggested DM thing, so this doesn't apply. Can you replace the damage type of animate objects with another damage type from my spellbook?