I don't use Outlook.com, and there's not a whole lot of coverage of that service around the forums here, and there are some reports of compatibility issues with the Apple Mail.app tool.
This discussion shows how to set up and verify alternate reply addresses with the Microsoft Outlook.com service. I'd start there, and see if that's what might be tripping up this alternate-reply sequence here; if there's a requirement to register addresses before use.
Failing that, there are a couple of ways to pick out the return address at the mail server, and it's quite possible that Outlook.com is picking up a different address, or not honoring the address being sent. That processing is much tougher to work around, as it involves changes by Microsoft and particularly the folks that operate Outlook.com servers.
Failing all that, your options aren't particularly good. Load Microsoft Windows and Outlook, and see if that works. Or load the Office mail client. Alternatively, migrate off of Outlook.com mail.
My general preference is to host mail through a domain that I have registered, and with a provider I've contracted with for service, as that makes porting mail services to another provider vastly easier, and it's easier to deal with issues and changes at the mail provider; whether that's bankruptcies, reorganizations, acquisitions or simple service cancellations. Free services can be costly. Paid providers are cheap, and mail client support is generally better, and the mobility of your mail to another service provider — as mentioned — is much better.