When you ask the question, where is the Catholic influence on the American Political System, the answer is not readily apparent.  Was not our country founded by Protestants?  And some of them had pretty insulting things to say about Catholics.
Well, there is more to it.

The Roots of American Order by Russell Kirk
Traces the Hebrew, Greek, Roman, and Christian influence on the American Constitution and American law.  Any Christian influence which goes back more than four or five hundred years is Catholic.

The Political Thought of St. Robert Bellarmine  
This is a strong argument that the political thought of St. Robert Bellarmine was formative in the founding documents of the United States.

The Law of Nations by Emmerich deVattel, 1848 translation should be available from your State Law Library. 
This book, once a basic of American legal studies is now out of print.  There is a 1916 translation which is less clear than the 1848 translation.  Translated from French.
This book is so anti-Catholic that it was placed on the Index of Forbidden Books.  It is not light reading.  But in fact, and for the good, it very strongly influenced the Founding Fathers and, except for its specific anti-Catholicism, contains the political thought of St. Robert Bellarmine through the lens of the Catholic-friendly philosopher Leibniz and his unfriendly disciple DeVattel.  I'm guessing that this volume is the source of much of the Bellarmine influence so visible in our founding documents.  Remember that many of the Founding Fathers were quite anti-Catholic and would not readily have received a "Papist" teaching.  Isn't God funny?  Never a door closes, but a window opens!

Tranquillitas Ordinis by George Weigel
A discussion of the just order of life according to traditional Catholic theology, -- including just war.
The Catholic Thread in Civic Thought