1762, "one who bounces," agent noun from bounce (v.), which originally meant "to thump, hit." Given various specific senses in 19c., such as "boaster, bully, braggart" (1833); also "large example of its kind" (1842); "enforcer of order in a bar or saloon" (1865, American English, originally colloquial).
Now it so happened that the brakeman was what is known, in the language of the road, as a "bouncer." That is, he was a hybrid combining the qualities of a brakeman and a bruiser, and was frequently called into requisition by the conductor to take the dirty work of ejecting tramps off of his hands. ["Staats," "A Tight Squeeze," 1879]
"The Bouncer" is merely the English "chucker out". When liberty verges on license and gaiety on wanton delirium, the Bouncer selects the gayest of the gay, and — bounces him! [London Daily News, July 26, 1883]