Need a "high-reputation" service for critical emails


New Email
One of my clients runs a service where some of the emails he sends to his customers are of such a highly critical nature that if a particular message does not arrive by a certain time, he may be liable for fines in the tens of thousands of dollars.

He's currently hosted at In Motion Hosting, and recently has had several bounces saying his messages will be delayed 24 hours, which is unacceptable. These bounces come with the "451 4.7.500 Server busy. Please try again later" error code. Researching that seems to indicate that it is the result of the Microsoft Office 365 service, more specifically from their anti-spam filters. So these bounces are caused by the remote end and the fact they're using the Microsoft email cloud, apparently.

The In Motion Hosting mail server is not blacklisted but Microsoft doesn't just allow those emails through every time for some reason. And In Motion tells me their email service is not intended for high-critical emails, they're primarily a web hosting company so I "may want to look elsewhere". Great.

So I need to find what I'm calling a "high-reputation" service which I can run the client's domain name on, and which can be run IMAP and SMTP using Outlook clients. Is there such a thing? Or another solution? Bounces of any kind are just unacceptable, though I realize the email system is not 100%. However, he has had occasions where even after multiple retries, he was simply unable to get an email delivered, and it was a critical one that caused a bit of a ruckus. He's let me know he's not in the mood for any more of that.

So I'm just looking to improve the odds of delivery I suppose. Any ideas?