I am not sure where the throughput being proportional to the square of the slit width comes from. There are trade offs between slit width, SNR and resolution but the change in throughput with slit width depends on various factors. Ignoring diffraction effects, for an extended source the increase would be proportional to the slit width but with a stellar source it depends on the FWHM of the star image relative to the slit width. A good guide is to aim for a slit width equal to the star image FWHM. With a slit wider than that you are gaining little in throughput but losing resolution. If the slit is set narrower you will increasingly lose more throughput but you may prefer to take the resulting increase in resolution if you still have enough signal to still give the required SNR.
Regardless of the camera or gain, shorter exposures means fewer photons and thus higher shot noise (which dominates at high SNR). There is no free lunch here