AF will work but will be slower, especially in less good light.
f8 instead of f5.6 means one step slower shutter speed or more image noise.
The longer focal length with TC increase the risk of shake (VR will have a more difficult job).
Including the D500 DX crop factor (1.5x) you already have 750mm focal length,
which to me seems like good starting point for bird photography.
TC:s can be useful in some situations,
but if buying a nice and sharp lens with fast AF,
I wouldn't put a TC on it the first thing I do...