The Bible does indicate that an intermediary can be used in a confrontation with an offender if. . .
- The offender is a believer.
- The offender will not listen to you in private.
The Bible does indicate that an incorrect use of intermediary is one that is. . .
- Out of order according to Jesus' teaching in Matt. 18:15-20, or the intermediary is used before a private encounter with the offender.
- One that is judged by unbelievers. A lawsuit would be a good example of this. 1 Cor. 6:1-8.
- One that is physically violent or spoken with harsh words. 1 Tim. 5:1-2, Matt. 5, etc.
In this answer, the following passages will be used:
- Matthew 18:15-20
- Matthew 5:38-40
- 1 Corinthians 5:11
- 1 Corinthians 6:1-8
- 1 Timothy 5:1-2, 19-20
I. Jesus' Teaching
You already mentioned Matthew 18 in your question, so you've likely read it. Jesus lines out a procedure of confrontation with a brother.
15 “If a believer does something wrong,[a] go, confront him when the two of you are alone. If he listens to you, you have won back that believer. 16 But if he does not listen, take one or two others with you so that every accusation may be verified by two or three witnesses. 17 If he ignores these witnesses, tell it to the community of believers. If he also ignores the community, deal with him as you would a heathen or a tax collector. 18 I can guarantee this truth: Whatever you imprison, God will imprison. And whatever you set free, God will set free. 19 “I can guarantee again that if two of you agree on anything here on earth, my Father in heaven will accept it. 20 Where two or three have come together in my name, I am there among them.”
Matthew 18:15-20 (GW)
Jesus' outline for confrontation with a brother:
- Confront the offender privately.
(You + Offender)
Move on to step 2 if the offender does not listen to you.
- Confront the offender privately with one or two additional people.
(You + Offender + Witness 1)
(You + Offender + Witness 1 + Witness 2)
Move on to step 3 if the offender does not listen to the witnesses.
- Bring the offense up to the gathering[b].
(You + Offender?[c] + Gathering)
If the offender does not listen to the gathering, treat the offender as a "heathen" or "tax collector".
From here, someone else may need to go in-depth about how a "heathen", or "unbeliever" and a "tax collector" was treated. A basic summary would be Paul's statement:
11 Now, what I meant was that you should not associate with people who call themselves brothers or sisters in the Christian faith but live in sexual sin, are greedy, worship false gods, use abusive language, get drunk, or are dishonest. Don’t eat with such people. 1 Corinthians 5:11 (GW)
To move on in the confrontation procedure, Jesus' Sermon on the Mount and classic teaching is used:
38 “You have heard that it was said, ‘An eye for an eye and a tooth for a tooth.’ 39 But I tell you not to oppose an evil person. If someone slaps you on your right cheek, turn your other cheek to him as well. 40 If someone wants to sue you in order to take your shirt, let him have your coat too."
Matthew 5:38-40 (GW)
Jesus calls His followers to be perfect. "Turning the other cheek" is a classic example of how Christians are expected to act. This would apply to both believers and unbelievers, but one must begin (and end) here if an unbeliever has committed an offence against them since an unbeliever skips the believer confrontation process.
II. Paul's Teaching
Paul teaches that a lawsuit against a fellow believer is "wrong" and "cheating" because the world does not judge people as God judges them. A man-made judicial system does not equal God's judgement. Even if a judge hearing the case was a believer, the judge would still be following man-made statutes, laws, rules, procedures, etc.
1 When one of you has a complaint against another, how dare you go to court to settle the matter in front of wicked people. Why don’t you settle it in front of God’s holy people? 2 Don’t you know that God’s people will judge the world? So if you’re going to judge the world, aren’t you capable of judging insignificant cases? 3 Don’t you know that we will judge angels, not to mention things in this life? 4 When you have cases dealing with this life, why do you allow people whom the church has a low opinion of to be your judges? 5 You should be ashamed of yourselves! Don’t you have at least one wise person who is able to settle disagreements between believers? 6 Instead, one believer goes to court against another believer, and this happens in front of unbelievers. 7 You are already totally defeated because you have lawsuits against each other. Why don’t you accept the fact that you have been wronged? Why don’t you accept that you have been cheated? 8 Instead, you do wrong and cheat, and you do this to other believers. 1 Corinthians 6:1-8
Paul's guidelines to Timothy support Jesus' teaching from Matt. 18 detailed above. If an offense is not corroborated by two to three witnesses, it should not even be considered.
19 Don’t pay attention to an accusation against a spiritual leader unless it is supported by two or three witnesses. 20 Reprimand those leaders who sin. Do it in front of everyone so that the other leaders will also be afraid. 1 Timothy 5:19-20
 "Brother" will be defined here as "a fellow Christian", or "one who claims to be a follower of Christ".
[a] Matthew 18:15 Some manuscripts and translations add “against you.”
This would need to be addressed in the Biblical Hermeneutics site rather than here if it was to come up as an issue. https://hermeneutics.stackexchange.com/
[b] Ecclasia -- the gathering, congregation, assembly, or church.
[c] It is not clear if the offender needs to be present while the offense is brought up to the "gathering".
Further: Paul gives guidelines on how to confront certain believers. Addressing Timothy, Paul advised him to discuss problems with fellow believers as if they were Timothy's family members. Though it does not answer the question, vs.1-2 help differentiate between different scenarios in which one might consider confronting an offender. Paul does seem to suggest gentle words with certain people and a reprimand with more influential spiritual leaders.
1 Never use harsh words when you correct an older man, but talk to him as if he were your father. Talk to younger men as if they were your brothers, 2 older women as if they were your mothers, and younger women as if they were your sisters, while keeping yourself morally pure. 20 Reprimand those leaders who sin. Do it in front of everyone so that the other leaders will also be afraid. 1 Timothy 5:1-2, 20