Hello Kelly,
Thank you for using Microsoft products and posting to the community.
We fully understand the inconvenience the issue has caused and that the current situation can be frustrating. But from the error message you shared, it seems that your boss is receiving email delivery failure notifications from the email address (mailer-***@googlemail.com).
Typically, if you use Gmail as your primary email address, you may notice a specific “mailer-daemon google mail” notification stating that your email could not be delivered to the address you placed. It helps you to know that your email won't reach the recipient you want it to. So, your boss is using the @gmail domain for his email address, right?
If that's the case, then it may be a better option to contact Gmail support directly to assist you in continuing to troubleshoot this issue. Because of our limited familiarity with Gmail as Microsoft support and the fact that this issue involves private account information, troubleshooting steps may need to be provided by specialized Gmail support. Below is the contact Gmail support I found for your reference.
Disclaimer: Microsoft provides no assurances and/or warranties, implied or otherwise, and is not responsible for the information you receive from the third-party linked sites or any support related to technology.
Your understanding is highly appreciated! I hope the above information is helpful.
Sincerely
Mavis - MSFT | Microsoft Community Support Specialist