The behavior of a speaker when pushed hard is to become nonlinear as it reaches the end of its play (movement) to keep the cone from self destructing. This is a very soft knee progressive ratio, instant attack and release wideband compression, which also provides additional musical 3rd harmonic distortion. A triode saturation in conjunction with a wideband compressor and a sneap style multiband comp before the IR will help bring some life into the cabinet.
There is more information about speaker distortion in the first link in my sig below, I talk about how a real speaker works. The more you understand how speakers distort, the better you can use vst or other hardware means to better emulate speaker distortion.