Forgive me for interfering in the detailed analysis of the doppler and related satellite engineering parameters, but there is a comment from Robin Clark to the effect that at apogee the satellite will have no "vertical motion".
I'm not sure which vertical is being referred to here, but assume that the author means the change in satellite altitude is zero. This seems correct to me, but that does not mean that the satellite will produce a non-zero doppler shift - there are x and y components to the motion as well as z - the satellite moves in a 3-dimensional figure of eight.
Apologies if the x and y components are irrelevant to this discussion - I don't pretend to understand the detail.