I use a workaround for that. I also have a GMail account, which I use for stuff like internet forums reply notifications and corporate spam which I have actually signed up for.
Instead of adding this low-priority email account as a separate inbox, I forward the email to my main Exchange account on Office 365, where email with that To: address is filed into a folder using rules. I think you can also get Exchange Online to pull the emails from Gmail, instead of forwarding them, but I have not ever used that feature.
That's what I do for a my personal Hotmail account as well, but the important thing to remember is basically there is no alert when the email is sorted into folders. You only get alerts for emails coming into the main inbox.
Because getting alerts for sorted emails is desirable when I am at my laptop or desktop computer, I use an Outlook add-on which allows that.