So your password manager uses your phone as 2FA, and the credentials inside your password manager also use your phone as 2FA? Hmmm…
So essentially, you can’t bring your phone, that’s the main issue. Does your authenticator on your phone support exporting a backup? Then store that in your password manager if that’s possible and set up an alternative 2FA for your password manager (SMS on the burner phone number perhaps or a security key). Then when you arrive, reinstall the authenticator on your burner phone and import the backup.
Perhaps you could also print an encrypted version of your Bitwarden TOTP secret on a QR code and bring it with you in your luggage?
So, encrypt the secret with a passphrase you can remember, encode the entire thing in a QR code and print it on a piece of paper. Easy.