> and provides no straightforward API for me to reconfigure its logger.

As far as I know, you just need to do something which configures a Flask logger, then call logging.config.dictConfig() with a suitable configuration which doesn't disable existing loggers and configures the Flask logger how you want. (The dictConfig call should replace any existing configuration with what you've passed to it). See this:

https://stackoverflow.com/questions/11816236/how-to-config-flask-app-logger-from-a-configure-file