I think "You, You, You" is the best choice from "The Visit" from a strictly musical/lyrical standpoint. However, "Yellow Shoes" shows the entire cast and it's upbeat (which is not true of most of the score).
Fun Home really should do "Welcome to Our House on Maple Avenue" (perhaps with some of the Prologue in there). "Welcome to Our House on Maple Avenue" is the music used in the advertising and that will jog people's memories. It also shows the family and that's probably a good thing. It's relatively upbeat and presents the dynamics of the story.
On the Twentieth SHOULD do "Babette".
On the Town-"New York, New York". No question...
The King and I-"Getting to Know You"-show the adorable children with Kelli O'Hara. (Alternatively, "Shall We Dance?")
Something Rotten-"Welcome to the Renaissance" (personally, I'd hold back on the show stopper-it's like giving the best part away (not that I think it's the best part but most people do) into "Will Power". (An alternative is "I Hate Shakespeare/Will Power" medley to show both Borle and D'Arcy James and the two sides of the story.)
An American in Paris-"Stairway to Paradise" with a huge dance break. (Because the dancing is the star of the show here.)