I never actually "solved" this, but I switched to Caprine and I'm no longer having any problems. It actually works much better than the official app (it lets you send audio files, etc).
This is a pretty good electron desktop app : https://sindresorhus.com/caprine/
It's a wrapper around messenger.com, with some extra features added. Source code here: https://github.com/sindresorhus/caprine