Resolving two nanosecond pulses with filter
The problem is that your low-pass IIR filter matches the original filter's magnitude response but will definitely not match its phase response.  

The IIR filter will have a kernel that is larger closer to $t=0$ than your plotted filter kernel.  This means that, even if the magnitude frequency response matches, the phase frequency response will not be similar.

So you have the filter's impulse response, which is sort of like a long FIR filter.

Unfortunately, designing an analog filter that matches this response is a little tricky. Most techniques for designing such filters assume you don't care about the phase response.  

One way might be to take your FIR current filter and perform model order reduction on it. See, for example, matlab's balred method that does balanced truncation (I think), though you may need to do some discrete-to-continuous or vice-versa transformation to get things to work.