Sounds like a great opportunity to convince them to switch to a proper email service! 
So back to your situation. Why not simply use their SMTP server for the outbound email? Do you demand to use webmail, or does domain X’s mail server? You can only accomplish this with a desktop or email client - or webmail client installed externally to MiaB.
There is no solution that is not an unsupported modification to MiaB. To accomplish your goal with the limitation you have you will need to configure postfix to handle that specific domain differently.
Explain why it is most, not all. Or are you the exception?
Probably the simplest suggestion is to host your personal emails on a subdomain which is hosted on the MiaB. Your email address will be me@kimusan.familydomain.tld. When an email is received by you and you need to reply to someone@familydomain.tld, then there should be no issue. This method also eliminates the ‘defective mail solution’ completely as far as you are concerned. This would require you to have access to the DNS records of the domain to set a NS record pointed to your MiaB. And the domain needs to be registered with a registrar whose DNS is smart enough to accomplish this…
And if changing your personal email address is too much of a hassle, continue to have mails sent to you@familydomain.tld - then forward them to you@kimusan.familydomain.tld and reply always from you@kimusan.familydomain.tld. Eventually you can make the old email address go away, or not. Basically, this should be your domain Y.