I thought it would, but as I mentioned, this happens even when the phone’s screen is on. The app isn’t killed (it has the lock icon toggled on just to be sure), the persistent notification is always showed even on the lock screen. I also have the battery optimization disabled in the developer settings (it’s a separate setting for MIUI alone). Basically all the settings from the link you’ve provided I already have set for K-9 Mail and the app have issues with PUSH.
I wanted to note that for AOSP based ROM I also had the same issue, so it’s not related only to MIUI.
BTW: I wanted to ask one thing. Should K-9 mail register for Google Cloud Messaging? I’m asking because when I was using AOSP with microG I haven’t seen K-9 mail on the app list.