bark, chirp, cry
啼 centers on vocal sounds made by animals or in distress: barking, chirping, or crying out.
啼 combines 口 (mouth) with 帝, which likely contributes the sound. The character expresses vocal sounds such as crying or animal calls.
A mouth 口 beside 帝, like an emperor's decree, but here the mouth is making animal sounds: barking, chirping, or crying.
For テイ, imagine a tea kettle whistling like a bird chirping: tea -> テイ, and the sound comes from the mouth 口.