I think it is a combination of the relative sizes of $v/c$ versus $v^2/c^2$ combined with the requirement to make sure a tangential velocity has no confusing radial component with a high level of precision throughout the measurement.
I suppose ideally you want something moving in a perfect circle and emitting radiation towards the centre. If the tangential velocity is $v_t$, then to isolate the transverse shift, you need to know any radial component to much better than $v_t^2/c^2$.
As an example. If I have an emitter moving at $3\times 10^4$ m/s, then the transverse Doppler effect is of order $10^{-8}$ compared with a "normal" Doppler effect of $10^{-4}$. Thus any radial motion must be eliminated (or characterised) to much better than 1 part in $10^4$ to isolate the transverse shift.