Middle English; earliest use found in Lay Subsidy Roll. From fire + beet + -er, in later use with remodelling of the second element after beater.
fire beater2
/ˈfʌɪə ˌbiːtə/
noun
An instrument for extinguishing a fire by beating it down repeatedly, typically consisting of a wide flap of leather or other material attached to a long handle.
Origin
Late 19th century; earliest use found in The Month: an illustrated magazine of literature, science and art. From fire + beater.