Yes, I agree that in this particular case it actually sounds more like "doin' it's not your style." What I actually hear is that there is just a kind of pause or gap there the "r" should be, "doin' it's not [?] style." There's something in the [?] but it could just as easily be a very soft "ch."
I assume, by the way, that you have some reason for thinking the lyrics DO say "Sinatra style?" The previous line about "serenade you" certainly makes it possible. But all of the Internet lyrics sites seem to be based on people listening, and are often inconsistent with each other (as well as being rendered with incorrect punctuation, etc.) Without going out and buying sheet music, there are few authoritative sources for lyrics!
Lyrics are extremely difficult because a) the singer often distorts the normal phonemes to express emotion or convey "style;" b) the song distorts the rhythms and intonation of normal speech; c) the language is often evocative and emotive, so it doesn't make logical sense and you can't guess from context.
One can't imagine a song lyric saying
"If Philip had been at the meeting, oh baby, he would have been able to present his plans, his sweet sweet contingency plans, for what he thinks we should be doing, shibbedy-bibbedy-sh-boo-bop, if we find out that they had already decided not to act, oh-woh-woooooh-woh!"
The fact that folksingers coined the word "mondegreens" is especially interesting because typically folksingers are solo singers backed by one instrument, singing words that actually tell a coherent story... and singing a soft voice, without the intentional vowel modifications of a trained vocalist. That is to say, the general level of diction is a lot better than in a popular song.