There is a trick. Sign up with them, configure custom domain, but don’t set your MX records. On your MTA, route mail that goes to Outlook/Hotmail to go through their MSA with your Outlook credentials and headers stripped as if submitted directly by user. Outlook gets mail through their own MSA it trusts, rest of the world talks to your mail system directly. Replies still go to your MXes.

I haven’t productionized this setup (I send maybe a dozen emails a year) but it worked for me ad hoc a couple years back, when I had to send a mail to an Outlook-hosted address and they blackholed everything I tried to send while 250 OKing me.