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 $v_t^2/c^2$.