Green’s Dictionary of Slang

1. a fool, a simpleton [? the bird’s willingness to be trained to sing].

1627		G. Peele Merrie Conceited Jests 24: He [...] takes this Bulfinch by the wrist, and carried him into the privy.
1629		R. Brome Northern Lasse II vi: What a Bulfinch is this! sure ’tis his language they call Bull-speaking.

2. a sovereign.

1825

C.M. Westmacott Eng. Spy I 129: I’ll bet a bullfinch that you know the place well enough.

3. (US) an ostentatious, showy individual.

1922

Wash. Times (DC) 21 Aug. 4/4: Funny, ain’t it, how a frill will get balmy over a bull-finch that is a past master at tea-room trots and parlor petting?