Neal Caffrey is a con artist, and therefore inherently slippery and difficult to pin down, but while I get the impression that there are some pretty diverse interpretations of Neal out there, not to mention portrayals in fan fiction, most of the time I just feel like the odd one out, the killjoy who insists on seeing Neal as the thief - the antiheroic individualist - first and foremost.
Cops will sometimes become crooks; it's one of the hazards of walking the line between legality and illegality. But professional crooks, even the nice ones like Neal (who are polite, and abhor guns, and would never kill anybody, or associate with those that do), are much less inclined to become conscientious, civic-minded peace officers. "Informant", maybe. Or "consultant on a short-term basis", for as long as he deems it pragmatic (to further his goal of finding Kate/catching her murderer; or because working with Peter is just too much fun), but Neal will never assimilate the call of duty that others might have. A respect for - and motivation to uphold - the law as an institution just isn't there.
What's right and fair is what Neal thinks it is at any given moment. He would never place himself, voluntarily and permanently, beneath the authority of an abstract body. The show is at such pains to bring home the fact that Neal does trust Peter - at a professional level, but more importantly at an interpersonal one - because the entire premise of the show hangs upon that trust. We can accept that a criminal like Neal would submit himself to the law so long as he is working under someone he believes that he can trust, someone who will stick his neck out for Neal even when it might not be professionally convenient to do so.
But the FBI? What reasons would Neal have to trust in the FBI, even before he learnt that it was involved in Operation: Mentor? None whatsoever. And why on earth would Neal, a fiercely independent operator, work for an organisation that he couldn't trust, unless he was getting something out of it? Something more than the warm, fuzzy feeling that he was "helping people"?
I've seen a lot of people complain about Kate being the McGuffin that brings Neal and Peter together, but let's not forget that she also keeps the working relationship together. Without her, Neal and Peter would still share a camaraderie, a liking and respect for one another, but I don't think for a minute that their friendship would be enough to keep Neal in White Collar Crimes, because it simply wouldn't sit with his beliefs and worldview. As a matter of fact, the only place in law enforcement I can see Neal maybe finding a permanent place in is Internal Affairs.