I’ve tried FairEMail and SimpleEmail, but they require opening the app to see emails. They seem to be forks or versions of each other without background syncing and (sorry) a terribly ugly interface. Neither seem to have filters.
Background syncing shouldn’t be constantly on, just every 5-10 minutes or so. Normally I check once a day, but forget to quite easily and then miss an email. If it checked automatically even twice daily, that’d be enough.
Your OS might be killing (outright or effectively) your app - check dontkillmyapp.com for possible solutions. Or the app doesn’t have notifications enabled at the app and system level.