My dog doesn’t come to my whistle because I’m bad at whistling.
A mouth flute... Normal flutes you blow air through to make a sound. How do you blow air through your mouth to make a sound? By whistling.
The reading is the kun'yomi readings put together. Thank 口 for being a body part, which tends to cause this to happen. So, you have 口 (くち) and 笛 (ふえ) put together, though ふえ changes to ぶえ, maybe because you can't whistle properly? Buuuweeee. Buwwweee.