idk if this works for all phones (I hope so!) but on my Pixel, you can long-press on a notification to open settings for that notification type. Changing it to "silent" will move it to the bottom of the list and no longer shows an icon in the status bar. You could also "hide" that notification channel entirely. I'm fairly sure this doesn't affect the ability to run background processes, but I could be mistaken
EDIT: This is a more recent feature so you might need the last one or two versions of Android, but if you have a phone with a notch I bet you're good :)
notifications silenced this way are not guaranteed to be kept in memory, so this roughly equals just stopping the app - especially on crapphones by Samsung and Xiaomi, which are the worst offenders of Android's background functionality.
EDIT: This is a more recent feature so you might need the last one or two versions of Android, but if you have a phone with a notch I bet you're good :)