But brother prosecutes his suit with brother, and that before unbelievers.

but brother with brother doth go to be judged, and this before unbelievers!

But a brother who has a cause at law against another takes it before Gentile judges.

Instead, brother contends against brother in court, and this before the unfaithful!

Must a follower sue a fellow follower? In front of unbelievers?

But brother - One Christian goes to law with another. This is designed as a reproof. This was wrong:
(1) Because they ought rather to take wrong and suffer themselves to be injured 1-Corinthians 6:7;
(2) Because they might have chosen some persons to settle the matter by arbitration without a formal trial; and,
(3) Because the civil constitution would have allowed them to have settled all their differences without a lawsuit.
Josephus says that the Romans (who were now masters of Corinth) permitted the Jews in foreign countries to decide private affairs, where nothing capital was in question, among themselves. And Dr. Lardner observes, that the Christians might have availed themselves of this permission to have settled their disputes in the same manner. Credibility, vol. 1:p. 165.

Brother goeth to law with brother - One Christian sues another at law! This is almost as great a scandal as can exist in a Christian society. Those in a religious community who will not submit to a proper arbitration, made by persons among themselves, should be expelled from the Church of God.

But brother goeth to law with brother,.... The relation meant is spiritual; it was usual for members of churches to be called brethren, they professing to be born again of the same Father, and belonging to the same family under Christ, the son, firstborn, and master of it: and a very wicked and shameful thing it was, that persons in such a relation, being of such a family, should go to law with one another at all:
and that before the unbelievers; which is an aggravation of their sin and folly. The apostle before calls them "unjust", now "infidels", such as had no faith in Christ, disbelieved the Messiah, and denied the whole Gospel, and therefore no faith or confidence should be put in them; for, generally speaking, such as have no faith, are not only wicked, but unreasonable men, men of no reason, conscience, justice, and equity; and therefore very improper persons for believers to bring their causes before.

But--emphatically answering the question in the end of 1-Corinthians 6:5 in the negative. Translate, "Nay," &c.