Breit-Wigner Peak - Not quite a Lorentzian?

The non-relativistic Breit-Wigner peak is given by: $$\sigma_{fi}=\frac{\pi\hbar^2}{q^2} \frac{2j+1}{(2S_1+1)(2S_2+1)} \frac{\Gamma_i \Gamma_f}{(E-Mc^2)^2+\Gamma^2/4}$$ where $q$ is the center of mass momentum of one of the particles and $E$ is the center of mass energy. From what I am aware $q$ is a function of $E$ (changing the center of mass energy will change the center of mass momentum) typically $q\propto E$ so we would get: $$\sigma_{fi}\propto \frac{1}{E^2} \frac{1}{(E-Mc^2)^2+\Gamma^2/4}$$ however in every source I can find (e.g. Martin, 2012 pg 27) it is stated that $$\sigma_{fi}\propto \frac{1}{(E-Mc^2)^2+\Gamma^2/4}$$ and is usually compared to been the same as a Lorentzian. My question is therefore what happens to the $E$ dependence of $q$ and why is it not considered?

Well, it is extremely accurate whenever $Mc^2\gg \Gamma$. Note that the distribution is almost entirely concentrated in the interval $$E=Mc^2 \pm {\rm few}\cdot \Gamma$$ because of the denominator that you kept in your last formula. But when $E$ belongs to this interval, $q$ may be well approximated by substituting $E=Mc^2$.
@LubosMotl answered the question already, I just wanna chip in some physical intuition for why $\Gamma$ is usually small compared to $Mc^2$. The Breit-Wigner-Formula applies to resonances, in fact only isolated resonances. In the case of overlapping peaks the more general Fano or shape resonances become relevant.