The Bat! needs its own app password to access your information. If you use your normal password or your Fastmail two step verification password on an external account, syncing to an external service won't work and you will see a password error.
Follow these instructions from Ritlabs on configuring your account for use with Fastmail. Enter your name, your Fastmail email address, app password and select if you're using POP or IMAP to fetch your mail. If you're using one of the Fastmail domains, The Bat! will automatically fill in all the other settings. If you're using a custom domain, enter the server and port settings manually.