Green’s Dictionary of Slang

(US tramp) a small-town sheriff or constable.

1926		J. Black You Can’t Win (2000) 163: In the language of the bums ‘yaffled’ is arrested, and the ‘town whittler’ is the constable, so-called because he is usually found sitting in some comfortable place whittling a stick.
1933		Ersine Und. and Prison Sl.
1944		C.B. Davis Rebellion of Leo McGuire (1953) 162: With the hue and cry on, there was too much chance of a town whittler picking a plum.