Guiliani was a Reagan democrat, he was never a genuine Republican. He ran against Dinkins twice; the first he lost and the second time he won only because Dinkins stunk so bad that even the most ardent base democrats had a tough time supporting Dinkins.
Was Guiliani a Rat Infiltrator to the GOP? Yes and no. He went from democrat (Carter) to independent (Ford) to republican (Reagan). He went to GOP only to further his career as prosecutor and because Reagan was so appealing.
Yes he was an infiltrator because he used the GOP brand in much the same way that Bloomberg used it, for his own ends. He was never a Republican in ideology, that he believed in limited government, in strict interpretation of the Constitution.
No he was not an infiltrator because of his appeal within the GOP as adhering to a philosophy of the Rule of Law.
So he is difficult to classify but I would say that under a Conservative Revolution of the type that rolls back the New Deal, rids the USA of the LBJ Great Society, discredits the Compassionate Conservatism of George W. Bush and seeks to appoint unequivocally conservative justices, I would say that Guiliani would not participate as a GOP ideologue, so that the GOP would leave him and he would be nowhere except as a New Yorker in NYC affairs.
That said, when he ran for President, one of his campaign promises, and one he was most likely to keep, was to appoint strict constructionists to the bench, though to be frank, on economic matters, he showed alot of conservative viewpoints, and he was in the hawk camp on foreign policy, but on social issues, he was to the left of Bill Clinton, and John McCain and everyone else (I could name a dozen democrats who are more conservative then him). I wouldn't mind seeing him as a republican senator or governor (for NY, and only NY), but on the federal level, only DHS, and thats it.