If the experiment is carried out all at the same gravitational potential level then there is no difference between SCC and SR/GR.
As I posted above Classical Doppler is not a 'relativistic mass' effect, it is a geometric effect caused by the relative movement; literally the photons are being 'squashed up' if the source is moving towards and are therefore observed at higher frequencies.
This is indicative of the frame dependence of energy levels. What is the zero point when measuring energies? To first order the relative motion shifts the frequency by an amount that is proportional to velocity. On top of his classical shift we factor in a relativistic correction, which does take time dilation into account.
(Here my LaTex has become all screwed up again