What lenses (focal lengths and minimum f-stops) would you plan to use with that larger sensor camera? Because roughly speaking, the "low light, high shutter speed" performance advantage of using a larger sensor relies on also using it sith a bigger lens, in the sense of one with a larger effective aperture diameter and thus larger front elements. And even then, the speed advantage comes with reduced DOF, maybe not good for your BIF scenario.
If you goal is to fill the same fraction of the frame with the bird, the focal length needs to be 1.6 times longer. To keep the same DOF, you then need to use an f-stop about 1.6 times higher — a bit over one stop. So to get the same shutter speed, you need to increase the ISO speed by the square of the f-stop increase, or 2.56 — a bit over two stops. How is the IQ ccomparison with that ISO speed difference?
If instead you plan to use the same ISO speed, the longer lens must offer same minimum f-stop at the longer focal length (like replacing the 70-200/2.8 by a 300/2.
and the ability to use the same ISO speed comes with DOF reduced by a factor of 1.6.
My gut feeling is that for wlildlife photography within the constraints on weight and cost (and DOF needs) that most of us are under, the smaller ILC formats serve better.
P.S. Given the far lower resolution of the Sony 7SII (12MP vs 20MP, and about half the "line per mm" on the sensor) getting equally detailed images ("equal pixel count on the bird") requires more than that 1.6x shift. Instead, if 12MP is enough, compare to the 12MP crops you could get with the 7D, with the crop allowing a shorter focal length than needed to fill the frame and so a lower f-stop for a given needed DOF. In that comparison, the 7SII would need twice the focal length (400/2.8 here we come!) and either four times the ISO speed for equal DOF, or half the DOF at equal ISO speed, or some balance like one stop higher ISO speed with "one stop less" DOF.
P. P. S. How do I disable smileys after the upgrade? I typed "parenthesis followed by comma" and it became an emoticon!