Keith poses an interesting hypothesis. I assume that the causal story for how bad weather reduces crime is that on extreme hot and extreme cold days that everyone (both good guys and bad guys) stay inside and thus “no trade” (i.e muggings) takes place.
Economists (as usual) have stepped up and done some “freaky” research related to this topic.  As discussed here, more video games = less crime. So, if bad weather keeps the bad guys inside and they hang out and play video games and then don’t commit crimes — we have a new logic chain on the urban quality of life benefits of climate change. Â
Now the “harvesting literature” would posit that such weather shocks will merely displace the violence to sunny days. This is testable but the police could anticipate this and try a little harder on those days while slacking off on the extreme nasty days.