Hibernate apps, but have them wake up periodically for a short time
Oh well, oh my … question almost included the answer. Still needs some testing (I've just set it up now), using tasker which has a Greenify plugin:


  
  Condition:
  
  
  Event › Display › Display Off
  
  Task:
  
  
  Task › Wait: 20 s (just in case)
  Task › Anchor: (Name, optional step)
  Plugin › Greenify › Hibernate: Pick the app
  Task › Wait: 30 min (or however long you want to have the app hibernated)
  Plugin › Greenify › Wakeup: Pick the app again¹
  Task › Goto: Action Number, 2
  
  Exit Task:
  
  
  Plugin › Greenify › Wakeup: Pick the app again¹
  
  


Note: That's how I would set it up. Had to skip the exit task as Tasker won't let me add one. So that's the crux here: I have to trust Tasker to restore the "previous state" correctly. If that doesn't work, I'll have to create a second profile for Event › Display: Display On (which would, being an EVENT, fire when the display will be turned on – it's not a state) and have a single task to do what otherwise the exit task would do.

Note 2: In order for this to work, you need to add the "target app" to Greenify's hibernation list first (i.e. tell Greenify it should hibernate it when turning the display off). If you don't do that, it won't ever be hibernated at all.



¹ if your app doesn't react on the wake-up command, see Xavier's answer for an alternative shell command to try