>> But, that assumes that the D600 at 3200 has an IQ that is better that of the D7000 at 800.
The short story is that the D600 loses by approximately 1/2 stop, given the assumptions in your quoted statement.
But there may be a better alternative "modest" cost lens in the form of a 300 f/4.
The long story...
Using this chart of Bill Claff's test results as a reference, and looking at the ISO 800-3200 range, which favors the D600 because it is relatively best compared to the D7000 in that range...
1. At the same ISO and DX crop, the D600 should beat the D7000 by about 1/2 stop.
2. The D600 with the full FX sensor will beat itself in a DX crop by about 1 stop.
3. The D600 with the full FX frame will beat the D7000 by 1.5 stops but you need to achieve the same focal length at the same focal ratio in order to do that. That makes that 1.5 stops very theoretical at the focal lengths under discussion.
It was previously suggested that a 300/2.8 would equalize things, at a price you are likely not willing to pay (based on your comments here). But that is not necessarily true because a 200/2 (using DX) is a bit less expensive and would still negate the FX advantage, but it would not negate the 1/2 stop inherent advantage of the D600 sensor verses the D7000.
Given the nature of the available lenses (300/2.8 vs 200/2), even if price is no object, an FX sensor at 300mm or greater cannot inherently beat an equivalent DX sensor in terms of noise. It can only equal it. Who beats who is a function of the age of the sensor in terms of development timeline.
What I am saying above is that a D600, with a 200/2 in DX crop mode should beat the D7000 (by 1/2 stop) using the same 200/2 lens in the same DX frame. But if Nikon replaced the D7000 tomorrow it would probably be a push because the inherent noise in an updated D7000 would be assumed to be about identical to the D600 in DX crop mode.
(Pixel density or resolution does not matter here because drilling down into the higher density sensor just adds noise. All my comparisons are based on un-cropped images at the same output size)
You are presenting us with a choice of a 200/2.8 lens verses a 300/5.6 lens, which is "not fair" because the 200/2.8 is a relatively better lens, by a full stop, at least when considered in the context of light gathering power (speed verses focal length). I understand that you are doing this because that is what you own, but the effect is to handicap the D600 by a full stop.
A better lens comparison would be a 300/4 against a 200/2.8, which would be more equivalent in terms of light gathering power and cost. Lens cost is more or less a function of light gathering power.
The Nikon 300/4 AFS, or a 3rd party equivalent if there is one, and the D600, would give you a 1/2 stop advantage over the D7000 + your proposed 200/2.8 although you give up the zoom which could be a problem. I don't think anyone makes (?) a constant aperture f/4 300mm zoom, which is what you would need to be fully comparable in speed and focal length range.
What I am trying to say here is that given the lens you have (verses 200/2.8) you are better off with the D7000D600 D7000 by 1/2 stop, purely in terms of noise. However the extra stop of depth of field provided by the 300mm f/5.6 zoom could work to your advantage in the case of group shots, and may be worth trading off a half stop of noise. If you only care about your daughter being fully in focus then that is immaterial.
(sorry about the double edits above- I managed to confuse myself at that point ).
(At 200/2.8 or 300/4 it can be very difficult to get more than one face in sharp focus, but it depends on the positioning of the subjects so it is impossible to generalize. Shooting around 200/5.6 now you may not have nearly as much of a problem)
The D7000 is well along in its product cycle so the idea of an update within the next year or so is not at all far fetched. That would be your least cost option and might be worth the wait, depending on your timeline.
As everyone else has said here, when you are reach-limited and cannot get close enough to take advantage of less expensive fast lenses, FX loses its advantage. And as focal length increases, it becomes increasingly difficult for FX to maintain an advantage because there is usually a 1 stop faster lens available (for DX) with 33% less focal length .
The only reason the D600 favors the D7000 in any case is the fact that it is newer and its sensor is apparently (per Bill's preliminary estimates) about a half stop better in terms of inherent noise. If Nikon had introduced an FX and DX camera on the same day then it is a safe assumption that the noise performance would be about the same, with both in DX mode.
As an afterthought... in principle a 300 f/4 prime lens should be less expensive than a 70-200 f/2.8 zoom, and considering only Nikkor lenses that is obvious. The problem here is finding a 3rd party 300/4 prime (or zoom) that competes with your 70-200/2.8 solution, and that may not be easy.
I don't know the 3rd party lenses well; I don't see a good solution on Sigma's product chart. Their "recently discontinued" chart lists a couple of 300/4 variants. I guess the market likes either slow f/5.6 zooms or the relatively very expensive 300/2.8 zoom, and not the f/4, which I think is the sweet spot in terms of price/performance.
An alternative is a 70-200/2.8 + 1.4x TC. That is optically almost identical to a 300/4 but gets into lens quality issues shooting whatever 3rd party TC you pick wide open. I don't have any opinion on that at all, having no experience with recent 3rd party TC's. I do have experience with the Nikkor equivalents.