Set up an embassy. Register your domain for Outlook, but don't really update the DNS (just add Outlook to SPF and DKIM to pass the validation, but don't change the MX). Then tell your MTA to send through Outlook servers when the destination is there (detecting this is a bit tricky), otherwise route normally.

I haven't really implemented this in production, but it worked for me one time as a proof-of-concept when I had an issue with disappearing mail - my message went through that time. Later it worked without any tricks, so I haven't bothered.

This sounds amazing. Know of any walkthroughs online?

Sorry, nope. It was my own idea (to best of my memory, I haven't seen this anywhere else, so I think it was an original one), I've tried it out, it worked, but I never finalized it, nor wrote anything about it.

It wasn't anything complicated, though. I've just did the documented steps to set up Outlook with my own domain (not sure if that's a free option, I have MS365 subscription for Office apps), except that I made no changes that would disrupt my existing mail system - I've added to SPF and DKIM instead of setting/replacing them, and I haven't touched any MX records at all. Then I've just grabbed Outlook's SMTP details and sent a test email to my other test Outlook account via SMTP and it got delivered with my email address, which gave me a confirmation that my idea had actually worked. I haven't really updated my MTA to do the routing thing, as I was about to replace it anyway (I did since then, replaced Postfix with Maddy).