They provided a schematic, but sadly they removed the values

You should be able to determine the component values using Octave or MATLAB. The circuit is a simple low pass filter. I'm sure the actual device is more complex.

However, what you want is:

B = G/A

where:

A is the Fourier transform of the output pulse

B is the Fourier transform of the impulse response of the desired pulse shaper

G is the Fourier transform of the desired output pulse

That will show you what frequencies you need to amplify and in turn the various component values for the Sallen-Key filters.

The Fourier transform of a narrow Gaussian in time is a broad Gaussian in frequency. The big limitation is how broadband an op amp are you prepared to pay for. The UPC2710 suggested by H&H is a ~$2 from Digikey.

Dump a bunch of pulse outputs to CSV format and post them. I won't make any promises, but it's an interesting question. I'm an instrumentation junkie, so I may well take a poke or two at it. Meanwhile familiarize yourself with Octave. It's free and very capable. And watch out for divide by zero in the equation above.

You should also look at the "Art of Electronics" by H & H. It includes a couple of photomultiplier circuits (cf pp 842-843) as well as an excellent section on filter design. The first circuit suggests using a 1 GHz BW amp such as an NEC UPC2710.

The filter you want is a low pass filter with a skirt that is approximately Gaussian. The transformation is apply gain and then filter. You need an integrator to provide the low frequency content of a Gaussian. A rough guess is that you'll wind up with a cascade of several filters with decreasing bandwidth.

It may well be that your current amplifier circuit is not broadband enough. I should also note that a Gaussian is a zero phase wavelet. That is not physically possible unless there is a delay. The real world is minimum phase. Systems do not produce output before T0.