Should be able to get confident step response with a small photodiode, with reverse bias; the kind of thing that's used for 100s MHz (or even GHz) in fiber. Not sure where to find such a thing (other than ordering one directly, possibly $$?).
Could compare step response with, like... atmospheric spark discharge, maybe? Xenon photoflash is too slow, some microseconds, but air sparks can be nanoseconds. Requires HV supply and a small amount of capacitance, of course. Keep good distance, partly just to attenuate the intensity of the light, but also to keep EMI away so you aren't measuring the spark itself, but its photo response.
I remember doing some experiments with this and getting good results with a generic IR photodiode, but I didn't have an IR LED at the time; I did demonstrate a xenon light curve (but again, at some us, that's not very impressive), and I think I did an air discharge but I don't seem to have a shot recording what it was.
Tim