Do you have access to an Android device (or the gmail app on IOS)? If you can sign that device into your Google account with one backup code you may be able to

Do you have access to an Android device (or the gmail app on IOS)? If you can sign that device into your Google account with one backup code you may be able to get Google login prompts without explicitly authorizing that feature in the blocked account settings page. I know when I have set up crappy tablets with my credentials they automatically started showing prompts to approve logins from other devices. If nothing else it may save you from running out of backup codes.