Peak Detector not capturing the peak
For high speed pulses, I would avoid amplifiers with fancy things in their feedback path.

Go back to the basic principles of how a diode or BJT work.  A diode or emitter follower driving a capacitor is a peak detector.  You can use a FET to reset the capacitor voltage to 0 to be ready for the next peak, or a resistor that bleeds off the voltage with a long enough time constant to not interfere with whatever you are trying to do.