Rome did not have a coherent political philosophy (probably not the best choice of terms, but let's work with it) by any means, and generally Romans didn't try to articulate one. Cicero did, but he was quite an odd duck. Even then there's nothing Roman that approaches something like Aristotle's or the Old Athenian's constitution of the Athenians.

Where Cicero most blatantly attempts this stuff is his De Republica and De Legibus, and even then these are consciously modeled on Greek works (Plato's Republic and Laws; a lost dialogue of Aristotle; superficially Polybius book 6). Cicero's most serious attempt to outline the Roman constitution was more a history of its development (with some theoretical underpinnings such as the corrupting nature of the sea for the moral fiber of a people, and the absence of foreign influence on Roman government) that a systematic account of its working or discussion about the aim of government. His model government was one in which the people awarded offices to the honorable, the Senate advised the magistrates wisely, and the magistrates commanded the people as necessary - a state of government Rome (Cicero wrote, speaking with the voice of Aemilianus) never sustained; rather the nobility and the magistrates constantly had to make allowances to the people to keep them in check or participating in the system, something Cicero's Aemilianus hates but recognizes as a necessity and so he decides not to discuss a perfect state but instead addresses the historical development of the state in practical terms - Rome is not the best state, but it is the best state possible given the circumstances. This is very different from the sort of ideal state Plato is babbling about.

If you wanted, you might try looking at Sallust; he at least seems to be theorizing about the corruption of the Roman state and it's origins - something about Rome getting its hands on foreign wealth and the loss of the metus hostilis - fear of the enemy. Sallust is damn hard to make heads or tails of though. Who his heroes are isn't clear, at least to me (the famous problem being whether we're meant to cheer for Caesar or Cato in the Catilinarian debate), and his villains look like heroes. Catiline, for god's sake, goes around congratulating his compatriots on their virtue and loyalty, speaks the language of opposing tyranny, and dies fighting ahead of his troops with all his wounds in the front. This is hardly Snidely Whiplash stuff. If he were anyone but Catiline he would look like the good guy.

The other place that occurs to me to look for "political philosophy" is in the mos maiorum, specifically how it was employed and contested in law, the Senate, and the courts. That could point you toward the pressure points at least - what did Romans disagree about that somehow had to be negotiated for public life to continue. Things like the separation of the military and civic spheres; the appeal against arbitrary judgment of magistrates, etc. seem like issues you could find addressed in practice if not explicitly theorized.

If you're interested, the Cambridge Companion to Greek and Roman Political Thought was published in 2012. I recently had my hands on it and was disappointed. It's heavy on the Greek side and probably takes on too many years to be very satisfactory for any particular period. Here's the BMCReview.

What do you mean by "the Founders"? Are you talking about the founding fathers of the USA?