If you use a different client, you can probably find one that will let you back up codes (I'm sure Authy supports TOTP). Also note that all you need for that backup is a "seed", which is a hash. IIRC Google Authenticator uses SQLite to store its data, I bet it's not hard to get the seeds out and back them up manually.

Things with backup or sync are risky, they add additional ways your authenticator can be compromised. There's an alternative that's similarly useful, but far more secure. When you add a 2FA code, print the QR code presented by the website and store it somewhere very secure in real world space. You can later scan it with any TOTP app, and it'll add the identical authenticator to another device.