Nominate a book that you think best encapsulates the American philosophical gestalt, but the catch is, it cannot be written by an American.
For the average workaday American I nominate Voltaire - Candide.
The Life and Opinions of Tristram Shandy, Gentleman by Sterne.
Life’s a joke, don’t take it too seriously.
O, how I wish that really met the OP’s criteria, instead of just being my dream. If you need me, I’ll be on my hobby-horse.
Can it be set in America?
If so I nominate The Loved Oneby Evelyn Waugh, a book set in 1948 L.A. but written by an English author.
I’m going to say no, because that would make it too easy. But people can mention it. 
Revision, it can’t be by an American or about an American.
This probably isn’t what you want, but there’s a 1940 novel by Christina Stead called The Man Who Loved Children. In recent years it’s moved up quite a bit in critical estimation and is now sometimes considered one of the greatest novels of the twentieth century. It’s the greatest novel ever written about Australian life that’s actually set in the U.S. Stead was an Australian writer who lived various places around the world during her life. During the time she was writing this novel she was living in Washington, D.C. Although the novel is clearly based on her childhood in Sydney, it’s set in Washington.