Green’s Dictionary of Slang

flat n.³

(US) a rejection.

1859

Bartlett Dict. Americanisms (2nd edn) 155: Flat [...] A rejection, dismissal.

In phrases

give someone the flat (v.)

(US) to turn down a suitor.

1866

F. Köhle Wörterbuch der Americanismen 54: She gave him the flat, He’s got the flat, She flatted him.