Detect "real" overcurrent/short-circuits vs. current peaks
The underlying problem description suggests a possibility of a low-tech solution rather than the high tech path. This is not to say that the mentioned integrator based approach won't work, of course.

Basically, how about two current limiting mechanisms in series: 


One that works well within the 40 microsecond time, but triggered only at close to the absolute maximum current rating for the affected components. This would have a sharp threshold, no RC or other integration involved, but would be auto-reset - the equivalent of a fast PPTC, from a circuit block perspective.
One that integrates threshold over time, either using an RC or an op-amp integration - hence it has a time constant potentially longer than the 40 microsecond limit. The threshold would be much lower than abs-max: Set it for the desired sustained-current limit.


This way, impulse currents such as for capacitor charging at start-up would get through, so long as they are under abs-max, while "slow-fuse" overcurrent protection would be in place for normal operating conditions.