EDIT: I suddenly got it to work by checking the "Require verification for browser integration" setting, setting up Biometrics in the Firefox extension, and then after logging in, unchecking the setting again. Now it works every time. Weird why this was necessary.
Sadly, the Fingerprint reading feature stopped working. It just says "Unable to set up biometrics
Action was canceled by the desktop
application", but the Desktop version is the latest, and it actually prompts me for a fingerprint scan. I don't know why this stopped working, but this was the one feature that was making Bitwarden worth it for me.