I've captured the two pulses for you. One rather small one of 140mV, the other 850mV. In the final version, the signal will be less noisier than in the scope shots, i basically used the voltage from a bench-top supply and put things on a breadboard with a not-professional coupled detector, so ignore the noises
That's very helpful thank you.
Without doing a fourier analysis on that, my MKI eyeball says that the leading edge of those pulses "fits" a sine wave of half period of about 3us. ie 6us full Period or about 160kHz. That means that high speed peak detector circuit should be able to keep up with that pulse (or very nearly so).
I am not sure what your constraints are, in terms of budget and time. I would do the following:
1. Make a protopye of that peak detect circuit. Feed your pulse through a unity gain inverting opamp circuit first (use a high BW/quality opamp, maybe the same type). I am thinking of a soldered circuit in your preferred format, not necessarily a PCB. You also need a mechanism for resetting the circuit between pulses obviously.
2. Then have a look on the scope. Can it follow? We think yes... Lets prove it.
3. At the same time, make some estimates of the inaccuracies of your aparatus / sensors. ie can you really get rid of that noise such that this is a 12-14bit signal? Get hold of a higher resolution scope if you can, to judge how clean this signal is. This is not that fast for a scope, so this should be possible. There is no point going nuts with 16bit ADC infrastructure if all you're going to measure is noise.
If the answer to 2 is "yes", then you no longer have a speed issue, and you can design your ADC stage for the accuracy which is justified by the answer to 3. (whether you need preamps / separate power supply etc).
I don't understand some of the other comments about RC / low pass filtering / estimate energy based on area under curve, etc. If I have understood your answer correctly, then you don't want to do that. I am taking what you said at face value. You want to measure that peak, as accurately as you can (just how accurate is yet to be determined).
Does that help?