I don't actually think the problem lies with Outlook but rather your ISPs. See many ISPs only allow users who belong to that particular ISP to use their outgoing email server. So when your are on ISP A and trying to reply to an email sent to your ISP B mail account, outlook will contact the outgoing email server for ISP B. But since your on ISP A, ISP B will not allow you to send mail through their server.
One solution would be to setup both email accounts to use a neutral server to send mail.
Gmail for example can be setup to send and recieve email in Outlook.
But all email sent through Gmail is rewritten so that it looks like it's send from your Gmail account. So for anyone receiving a mail sent this way, the mail will have a Gmail email address in the 'from' field instead of one of the two email addresses you have now.
But maybe someone else knows a better solution or an outgoing mail server that allows you to do this without it changing the email address.