I think perhaps if you created an SPF record that authorized your ISP to send mail on your behalf you might be ok.  It's at least worth a try.  You could also ask your ISP to try sending them an email to see if it goes through, if not then it is your ISP's problem perhaps. 

Hi,

We've got a problem when sending mails to one special e-mail recipient. This email is being returned with following error message:

SMTP error from remote server after RCPT command:
host: mx.destination_domain.com
TLS client certificate subject does not match sender domain

FYI: Mercury is not working as our direct mx server. MercuryC is forwarding all of our mails to the mail server of our ISP who is sending the mails to the world. And in general all is working properly except for that one recipient. We are not using any PGP encryption or something like that, but the submission to the provider is set to SSL with STARTTLS. The provider is accepting the mail, only the recipient's mail server is rejecting the mail.

I'm not really experienced in encryption and certifice handling. I'm only interested whether the problem is on our side or not.

Greetings

Joerg