After you setup Rely connector, change IP address of old Exchange 2003 server, and add that IP address to Exchange 2010 server. Change binding for relay connector to use that ip. That way you do not have to go to every client
and change setting for new relay server.
When you type get message This is probably not the site you are looking for! look what URL / certificate is used, and exactly which IP it is resolved.
But I think, that you forget to set certificate for services Enable-ExchangeCertificate -Server ExchangeCAS -Services 'SMTP','IMAP','POP','IIS' -Thumbprint 'A57D08CF688F0D4D157616EFD041E01B8BA69F23' -Confirm:$false
But be carful, where you run that command because certificate and services, can be set for remote server, and you install and set service url on wrong server.