AirPods don't work when connected to external monitor - sometimes
After weeks of poking at this, I've managed to figure it out.

This actually had nothing to do with the AirPods or the external monitor, rather the problem was that Google Hangouts has its own settings to choose which devices are being used for input/output that overrides the system settings. So, while I was frantically trying to change which device was supplying in/out in the system preferences, Hangouts was happily sitting on the devices it had chosen, ignoring those changes.

Once I realized what was going on, changing the setting manually in Hangouts from my other wireless headphones to the AirPods proves effective every time.