Is there a pattern in prases of Snuggly the Crow in the Dark Souls 3?

Each item has a pre-defined cry. Only items that makes it say "Pickle Pee" will give you the "Call Over" gesture.

Source: DS3 wikidot

I found a table with the information you are looking for on the Wiki.

Drop shows what item you must drop (not discard) on the nest with Snuggly. Receive is what item Snuggly gives you once the dialog has concluded. Response is the response Snuggly will give you once the item has been dropped on the nest.

Drop: Firebomb or Rope Firebomb.
Receive: Large Titanite Shard.
Response: Pump-a-rum.

Drop: Black Firebomb or Rope Black Firebomb.
Receive: Titanite Chunk.
Response: Pump-a-rum.

Drop: Prism Stone.
Receive: Twinkling Titanite.
Response: Pump-a-rum.

Drop: Loretta's Bone.
Receive: Ring of Sacrifice.
Response: Pickle pee.

Drop: Avelyn.
Receive: Titanite Scale x3.
Response: Pickle pee.

Drop: Coiled Sword Fragment.
Receive: Titanite Slab.
Response: Pickle pee.

Drop: Lightning Urn.
Receive: Iron Helm.
Response: Pump-a-rum.

Drop: Homeward Bone.
Receive: Iron Bracelets.
Response: Pickle pee.

Drop: Seed of a Giant Tree.
Receive: Iron Leggings.
Response: Pump-a-rum.

Drop: Siegbrau.
Receive: Armor of the Sun.
Response: Pump-a-rum.

Drop: Vertebra Shackle.
Receive: Lucatiel mask.
Response: Pickle pee.

Drop: Divine Blessing.
Receive: Very Good Carving.
Response: Pickle pee.

Drop: Hidden Blessing.
Receive: Thank You Carving.
Response: Pickle pee.

Drop: Alluring Skull.
Receive: Hello Carving.
Response: Pump-a-rum.

Drop: Undead Bone Shard.
Receive: Porcine Shield.
Response: Pickle pee.

Drop: Any Sacred Chime.
Receive: Help Me Carving.
Response: Pickle pee.

Drop: Shriving Stone.
Receive: I'm Sorry Carving.
Response: Pickle pee.

Drop: Xanthous Crown.
Receive: Lightning Gem.
Response: Pump-a-rum.

Drop: Mendicant's Staff.
Receive: Sunlight Shield.
Response: Pickle pee.

Drop: Blacksmith Hammer.
Receive: Titanite Scale.
Response: Pump-a-rum.

Drop: Large Leather Shield.
Receive: Twinkling Titanite.
Response: Pump-a-rum.

Drop: Moaning Shield.
Receive: Blessed Gem.
Response: Pump-a-rum.

Drop: Eleonora.
Receive: Hollow Gem.
Response: Pickle pee.

I hope this helps!