Pulse Doppler Radar: how to filter its Range-Doppler matrix?
Two suggestions that may independently, or combined, help to solve your problem.


Consider using a different window/weighting function for your FFT processing.  There are window functions with low uniform sidelobes such as the Chebyshev window. This window function allows for selection of a constant sideline level response.
It sounds like you are using a fixed detection threshold, consider using an adaptive detection threshold so that the threshold can vary with the combination of background noise and sidelobe artifacts.  Mathworks has a pretty good description for a simple case at Constant False Alarm Rate (CFAR) Detection.