smooth noisy irregularly spaced data containing peaks
The idea of cascading a non-linear then a linear filter is sound in the presence of spikes. 

Here are some observations which, if correct, could be exploited for better processing:


Data is circularly arranged, and in fact 2D
Spikes point outward,
The remaining noise seems bounded (the thick blue ring),
The angular sampling is dense, and could be affected by noise.


Some suggestions: 


Choose a novel angular sampling $\delta \phi$ that could be used for several datasets. It could be parametrized by a modelling of histograms with a one-sided exponential law, like $a\exp(-\lambda t)$, $t\ge 0$. 
Choose an angular width $\Delta \phi$, that will be used as the span of a 2D filter. 


Their combination would define a series of overlapping windows on which you would process 2D point clouds.

Inside each angular region defined by $[n\delta \phi - \Delta \phi/2, n\delta \phi + \Delta \phi/2]$, you can for instance:


define a nonlinear filter to eliminate strong spikes: for instance a 2D median, or a 2D weighted median (A fast implementation of two-dimensional weighted median filters). If you use the outward direction of spikes, an order filter, or a rejection of the top $x$% of the radius could be used. The idea behind weighting is to penalize the points that are the farer away from the angular sampling. A bilateral filter for point clouds could be an option too. 


Introducing a weighting in a median filter can bridge the gap between pure median and smoothing filters, and kill two birds with one stone: remove spike, reduce noise. As a result, this could suffice. If not, you can then:


perform a 2D linear filtering on the remaining points, in the shape of a center of mass allocation. 


One advantage with resampling to a regular angular grid is potentially to allow easiest further processing, comparisons, etc. And using 2D filters, you naturally follow the circular symmetry of your data.