If Josiah Forsyth had any real-world equivalent, it was the actor John Sewell Folds, better known by his stage name of "Jack Langrishe."
This file appears in: Spokane's "Mayor for a Day" (Who Never Was)

Langrishe was well known for his Shakespearean roles and was said to have played in every town in the west during his long career. Like Forsyth, he was known for his 'Hamlet.' Like Forsyth, he performed in Spokane in 1883 to great acclaim. Unlike Forsyth, he did not drunkenly fall into the Spokane River and drown. Langrishe retired from acting in 1885 and settled in Coeur d'Alene. He was elected as an Idaho state senator in 1890.
This file appears in: Spokane's "Mayor for a Day" (Who Never Was)
Spokane's "Mayor for a Day" (Who Never Was)
The little-known (and utterly fictional) tale of Spokane's "mayor for a day," Josiah Forsyth.