Is this a Gmail account or a custom domain? One idea, although it may be a long shot. If it is a custom domain on Google, you may be able to convert the account to a business account. If it is a regular gmail account, you may be able to signup for a google business account with a custom domain (you can do a 30-day free trial, although you may have to pay $5, or at least put a credit card on file, to gain access to the free support).

Either way, with a business account you will then have access to live Google phone support for free (which is actually pretty good from my experience). Call them and tell them you want to move your mail from your old gmail account to your new business account, and use their support to see if they can help you gain access. Again, not sure if it will work or not, but at least it would give you a direct line to their support teams at little to no cost. Then once you have shut everything down, or have completed everything you what you want to do, just close the business account.