I had never heard this variant on "nipping at one's heels," but it seemed so natural and plausible, I thought for sure it would be in the Eggcorn Database. But no; it's not there, and though the nibbling version gets a handful of Google hits, it doesn't come close to rivaling the standard nipping idiom.
That suits my sense of the words; I think of nipping (at someone's heels) as aggressive behavior, like herding sheep or chasing prey. Nibbling, on the other hand, means eating in tiny increments, or at least pretending to eat, and isn't a hostile act. The OED definition: "To take a small bite, or a series of small bites, at or from (a thing); to bite away gradually; to bite tentatively, delicately, playfully, or amorously."
There has been, historically, some overlap in the verbs' senses -- but not in the direction of the Ron Paul substitution. It's nip, which implies a a pinching action, that has sometimes been used as a synonym for nibble, according the the OED:
The milkcows were nipping the clovery parks (1839).
That durn brute was shakin' his ears and nippin' grass unconcerned as a can o' green corn. (1900)
Apparently, though, there have been moments when a writer felt the need of both verbs at once. And because one of those writers was George Meredith,* the OED has also recorded the "rare" word nibblenip, meaning, naturally, "to nibble and nip." The illustrative quote, from a poem of Meredith's (1883): "Haggard Wisdom, stately once, Leers fantastical and trips: Allegory drums the sconce, Impiousness nibblenips."
Maybe that's the word Ron Paul was looking for: "We're nibblenipping at his heels!" It does have a certain ring to it.
*The quoted lines will hardly recommend him, but Meredith wrote (among many other things) a comic-romantic-feminist novel every bit as deserving of a miniseries as "Jane Eyre" or "Sense and Sensibility." "Downton Abbey" is all very well, but every year I'm amazed that there's still no screen version of "The Egoist."