@jasty It has to do with the nature of SCP. Each containment unit's personnel are made to know as little as possible as each other. In addition, when you're paralyzed, you can't speak, the D-Class probably aren't trained enough to capture it blindfolded, and they're unwilling to bomb or kill the SCP due to loss. While the Oversimplified SCP makes the cockatrice look cute, it's a two-meter long lizard that rapidly spawns children, who grow in a matter of hours.
@Kneel_Breen It actually doesn't affect the nose, eyes, mouth, or any other major mucous membrane. The cockatrice prefers to eat into your body through them.