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.
The problem with this setup is that if I want to send email to fred@example.com and example.com is actually handled by Outlook, I need to somehow know beforehand that email to example.com needs that special routing. Or do you just route ALL your outbound email through Outlooks MSA?
You can usually figure that out by checking the MX records (though some spam filtering middlemen might make that hard to detect) and TXT records (if a domain imports Outlook's SPF list, it's probably on Outlook in the back).
Pretty easy to tell whether example.com uses outlook; it'll have MX records pointing there.
Great idea