Good question. I have a vague idea about how errors like this are catered for so I'll take a shot at answering your question. I stand to be corrected by anyone who's closer to SNe Ia cosmology. The short answer is that the discovery of dark energy is based on empirical calibrations, so any scatter in the progenitors of the supernovae is already accommodated.
The first thing is that the theory of Type Ia's is actually not all that clear. The overall picture of nuclear detonation in a carbon-oxygen white dwarf is pretty solid but there are some tricky bits to explain. For example, the accretion onto the WD must be within a very narrow range. Too little and the material builds up, surface detonations blast off the material, and the WD doesn't grow. Too much and, again, surface detonations wreck the picture. I think there's still a mismatch in the predictions from theoretical populations and the number of Ia's we actually see. Also, there's a lingering suspicion that there are other progenitor channels. There are already some underluminous Type Ia's and it is thought that some WD mergers will also cause trouble.
So, at the end of the day, the calibrations that are used for the cosmological results are empirical, so they already accommodate statistical scatter in the Type Ia sample without knowing whether or not we're accidentally confusing more than one type of event. The new spin-up/spin-down paper might explain some of the brightness variation in the samples but the spread is implicitly already in the dark energy result.
Refining our understanding of Type Ia's is big business for exactly this reason. As far as I know, the biggest contributor on the error of the pure SNe results is from the lightcurve fits. Improving those would therefore improve the precision of the cosmological result. But, that said, we now know independently that dark energy is out there. Dark energy was discovered before WMAP and those results are now bolstered by baryon acoustic oscillations. Those two results each centre on the standard cosmology, but with much bigger scatter. Fortunately, when we combine the data of SNe Ia, WMAP, and BAO, they all neatly converge on one set of parameters, which is why it is sometime called the "concordance" model.