I don't receive push notifications from Mail

I usually receive push notifications from all apps without issue. The one exception to this is Apple's Mail app and - strangely - only one email address. Push notifications, as well as notification badges, don't appear until I open the app, which defeats their purpose. I might've already found the reason for the problem but it creates more questions than it answers.

I use the Mail app for two email addresses: iCloud and my business email. On my iCloud address, push notifications and badges work just fine. On my business email the issues are as described above.

Now when I go to Mail > Accounts > Fetch New Data, I can see that the schedule is set to Push on my iCloud address. However, on my affected address it's set to Fetch. I wanted to change this one to Push as well but the option wasn't even available. I can only choose between Fetch and Manual. Is there a way I can fix this? Did I maybe make a mistake when adding the email address to the Mail app?

This is a painful issue because this email address is super important for me.

Thanks in advance to anyone trying to help.