So, I finally figured out why Weather Now kept struggling: I deployed it to a too-small App Service Plan for the load it was getting. And since most of the load is coming from bots, I hope that the robots.txt file I finally deployed properly will get them to stop.
I fixed the mysterious exceptions being thrown, too. But it turned out, the problem was simply load. A bigger App Service Plan brought the effective CPU from 100% to 28% immediately. And it'll only cost another $66 per month...
I'll experiment with other ASP sizes later this week. At least the app works again!