I have several of these images of an eagle flying ... (
J.K. May I offer the following?
When I go 'bird shooting' with my Canon and a 100/400, I choose a lower f/stop, then a 'fixed shutter speed' such as your 1/1600, depending on the bird at hand, and then I let the ISO 'float'... Many times, I will choose a f/stop just 1 stop above the minimum of the given lens I am shooting.... Yes, Yes, I know... Every lens has it's 'optimum sweet spot' and I understand that... but hear me out.
For shots such as this, your f/stop could be set further down, say f/6.7 (or lower) and that would allow your ISO to be up-scaled giving the shutter speeds remains @ 1/1600? Here is the logic behind that.
With this shot you have, I am going to 'make a general assumption' that your distance to the subject is 500ft... I know it could be further, but we will assume that for now...
Using a 'Depth of Field' calculator, and plugging in the numbers of f/6.7, 100-400mm lens @ 400mm, your given angle of view will give you a depth of fields of 199ft... that would 80ft in front of the subject and 118ft to the rear of the subject... and ... OF Course, you 'objective' is to be actually 'in focus' somewhere on the bird...
Given all of the above, and with an Auto ISO, the ISO range is going to be much lower than your given amount, and therefore, the 'Noise' will be reduced...
For the calculations I have given, please see the attached JPG, which can be calculated for any of the numbers using the URL Link below...https://www.photopills.com/calculators/dof
Hope this is helpful in some way...
PS... Given all of the above, yes you want to consider all elements of what would improve your photography but, having said that I would suggest that starting with a better image out of the camera would be the first solution and then add denoise programs on top of that
... After the above, you can also continue with the 'refinement' of your product with any of the many 'de-noise' programs on the market... Yes, there are some good one.