Green’s Dictionary of Slang

an idiot, esp. a weakling.

1954		Hepster’s Dict. 5: Hey, nosebleed! – hey, stupid!
1990		(con. early 1950s) J. Ellroy L.A. Confidential 13: He’s a nosebleed, and you can bet he don’t approve of on-duty imbibing.
2023		(con. 1962) J. Ellroy Enchanters 9: Gloryhound cop and all-around nosebleed.