Yes, Lincoln would have done "Between Two Ferns"

Many of his stories reflected his roots as a salt-of-the-earth rail-splitting everyman—an image, by the way, that icon was careful to cultivate. Historian Harold Holzer has written: “Lincoln’s admirers loved his down-to-earth style and earthy way with a comic tale.” And not all of his stories were fit for family consumption, In the movie Lincoln, the president regales his friends with what is euphemistically labeled “scatological” humor (spotting a picture in the bathroom of a home during a visit to London, Lincoln tells his British host he understands because “nothing makes an Englishman shit quicker than the sight of George Washington.”

Lincoln’s humor, as Holzer notes, was not an unalloyed political asset. “Foes leaped on such qualities as evidence of Lincoln’s coarseness and lack of dignity.” A carton published during the Civil War shows “Columbia” a symbol of the nation then as familiar as Uncle Sam—demanding the return of her “500,000 dead,” and Lincoln responds by saying: “That reminds me of a funny story.”

Indeed. Presidents and potential presidents have often been accused of crossing some line of propriety. Campaigning itself was once thought of as beneath the dignity of the land. In 1860, the Jonesboro (Illinois) Gazette scorned Senator Stephen Douglas for “going about peddling his opinions as a tin man peddles his wares…small business it is for a candidate for the presidency to be strolling around the country begging for votes like a town constable.”