Green’s Dictionary of Slang

(orig. US black) to argue, to answer back, to be cheeky, to talk excessively, to boast.

1962		P. Crump Burn, Killer, Burn! 67: Next time wait until you get those pounds before you start fat-mouthing.
1978		H. Selby Jr Requiem for a Dream (1987) 38: Ah doan mind him fat mouthin that [TV] set. [Ibid.] 176: He sittin back fat mouthin while he gettin ready to do some more time.
1980		E. Folb Runnin’ Down Some Lines 44: Lame talk refers to bad talk, in the sense of provocative or belligerent dialog. Expressions like smart mouth, fat mouth.
1999		E. Bunker Mr Blue 197: He was just fat-mouthing.
2006		posting at Seattle Post-Intelligencer 30 June 🌐 I know this stuff because [...] [I] was in a position to accurately know what the story was as opposed to your ignorant, clueless fatmouthing.