I understand that you hesitate to sent us the mail, but unfortunately in that case, we cannot help you.
Since Content-ID is missing in the eml file, there is no way to correctly link the resource from HTML body.
We wanted to check where the email got corrupted. But, if you have no access to the original Gmail message, you will hardly to diagnose it.
The only reasonable solution how to analyze the issue is to monitor incoming raw data to your mail server and check whether the received mail has Content-ID filled.
I got an idea to probably the simplest solution: ask your customer whether he can see the image correctly in the Sent Mails on his Gmail account.