I believe this is a false attribution. The phrase "rubbish heap of history" appears to be a translation of the German phrase "ein Scherbenberg der Weltgeschichte" ("a mountain of shards of history"), as translated by Annie Hamilton in her translation of the German historian Ferdinand Gregorovius' History of the City of Rome in the Middle Ages. It comes from a chapter of Gregorovius describing Rome's decline before Petrarch visits it, so is at least tangentially related.

What I think Hibbert is actually doing is putting the idiom "rubbish heap of history" into quotation marks to describe Rome in his own words; note that the quotation is divorced from the other Petrarch that he mentions on that page of the text (if you've looked at the Google Books version), and he is similarly ambiguous when he does the same thing again in The Borgias 10 years later.

I will admit to only skimming through a few of Petrarch's letters to check them for this kind of description in Latin, but although he frequently comments on the state of Rome and its decline, I do not believe this is something he ever wrote.