Maybe someone who knows more than me can clarify, but in my theoretical understanding Impulse Responeses are only going to be meaningful for linear systems. Eg, room acoustics are probably pretty linear: if 10% of the sound bounces back from a surface, it doesn't matter (within reason) if that's 10% of 60dB or 10% of 110dB.
If I'm interested in reproducing a speaker, they're only linear during the "boring" bit of their operating range. If they're operating linearly they're accurately reproducing the input sound (subject to EQ - which essentially what an IR is). If the speaker is colouring my sound in "interesting" ways, then that's probably because I'm driving it near its limit where it becomes non-linear - where they start to break up, and do weird things that can't be modelled by a simple IR. This isn't something most of us do - its more of a guitar thing, but also specific styles of bass playing where break up, and the speaker response are more important than wanting a big loud clean sound.
It would be possible to measure the IR at different sound levels by playing different levels of white noise (white noise is used for measuring rather than impulses, as impulses do bad things to speakers), but that doesnt' account for what would happen if a loud bass note is played at the same time as a quiet higher note. Otherwise all you're doing is applying an EQ to the output to (basically) roll off the high and low end, and add a peak around the frequency of the high pass filter.
I've not really used IR's and my technical understanding is based on a purely theoretical background (from a long time ago in a different field), so this is as much a question as a comment - how do IR's handle non-linearity? I can understand that speaker models can certainly be build that incorporate it but they wouldn't be IR's as I understand them.