He/she has no discretion. There are only a very few technical grounds. If the criminal is already serving time in the state of residence, he doesn't have to be surrendered until his sentence is finished. There was a late 1800s SCOTUS case to the effect that federal courts couldn't enforce it, which made no sense. In 1987, in a landmark case which name escapes me, the court reversed and held that the federal courts could enforce it. That ended about a century of its being, for practical purposes, discretionary...Do you read that as an obligation without discretion as I do or would something else give the governor authority to defy this statute?
941.02 Fugitives from justice; duty of Governor.—Subject to the provisions of this chapter, the provisions of the Constitution of the United States controlling, and any and all Acts of Congress enacted in pursuance thereof, it is the duty of the Governor of this state to have arrested and delivered up to the executive authority of any other state of the United States any person charged in that state with treason, felony, or other crime, who has fled from justice and is found in this state.