It's hard for me because, a) I'm not generally a portrait/people photographer, and b) I've got no real familiarity with Canon color profiles. I do know that I've seen more than a few side-by-side Nikon v. Canon body challenges that will speak to the glory of Canon skin tones, while praising Nikon for landscape detail and color depth, etc. I honestly don't know if you'll ever be able to get exactly what you want out of the D600 because, well, it is what it is. I started looking at the color profiling software, and the thing it does best is to get all the cameras to look the same in the same lighting conditions. But heck, when do you really get that? Obviously you can play with a basic profile and learn what hues you need to tweak for saturation in order to achieve a Canon-like profile, but then you'd need to do that for profiles under every possible lighting condition.
I hate to say it, but if I'm going to be consistent with my mantra use the right tool for right job, then I'm wondering if you might not just be better switching? If the Canon color profile is that important to you and your vision then why not bite the bullet and make the change? You're unfortunately heavily invested in a camera that has seen its market value plummet, so swapping for equivalent Canon gear will be costly. But if it's that important then it may be where you need to go.
Otherwise it may be time to start scouring non-brand specific, post-processing focused forums and asking the question, "How can I achieve a Canon-like color profile on my Nikon D600 without switching brands?", while ignoring the trolls who will slam you for having the wrong brand in the first place.
I've got the question queued up to my brother, and I'll let you know what he says.