OpAmp based constant current source/sink produces high peak current when load voltage is pulsed
The output capacitance of the IRFH6200 is 2.89 nF with 10 volts from drain to source. With zero volts from drain to source this capacitance will be in the realm of 10 nF so, when you apply the pulse to VL, irrespective of how long or short it takes for the control system to recover, you are (in effect) applying that same pulse directly to RL via a capacitor that starts at about 10 nF and diminishes to maybe 2 nF when the peak of the pulse is reached.


  What happens is, that there is a large unrestricted peak current
  flowing through RL for about 20µs


That initial current flow is due to the capacitor and the fact that the MOSFET, prior to the injection of the voltage pulse is hard-on due to the op-amps. How long it takes for the control loop to stabilize is down to choosing a faster op-amps (in part) but also it's down to being able to remove charge from the MOSFET gate (power drivers are usually rated in amps to accomplish this).

The gate source capacitance is about 10 nF and a measly 20 mA from an op-amp is going to reduce the gate voltage at a rate of 20 mA/10 nF = 2 volts per microsecond - it might take 5 to 8 us just to turn the MOSFET off.