You require more than the New Testament does when it comes to refuting false teachers and false teachings.
Paul and Peter did it publicly.
I understand that you are battle-weary but the enemies attack is relentless...Thus, the fight must press on. We have to stand firm and contend for the faith.
Publicly, yes, but I still believe it needs to be done in person, directly to them. God tells us if someone is caught in sin to go directly to them, if they don't listen, bring another and if they still don't listen, they're to be taken out of the church. We don't do that anymore. We talk about people behind their backs which doesn't do them or us one bit of good. Better to pray and ask God to reveal their heresy. He will.