We appreciate your thoughtful question and we want to expound on this together as we seek God's wisdom.
You are correct that Scripture holds both truths in tension. We are commanded to pursue unity, love, and peace with one another, and we are also commanded to speak the truth in love and to exhort and correct one another when needed. These are not contradictions but two parts of godly fellowship.
We are told in Colossians 3:13 to bear with one another, and forgive each other, if any man has a complaint against any, and in Ephesians 4:2-3 to walk with all lowliness and humility, with patience, bearing with one another in love, being eager to keep the unity of the Spirit in the bond of peace. At the same time we are told in Ephesians 4:15 to speak truth in love, in Galatians 6:1 that if a man is caught in some fault, you who are spiritual must restore such a one in a spirit of gentleness, and in 2 Timothy 4:2 to reprove, rebuke, and exhort with all patience and teaching. Proverbs 27:5-6 reminds us that better is open rebuke than hidden love, and faithful are the wounds of a friend. So love does not mean we avoid correction, but that correction must be done in humility, gentleness, and love, not in pride or harshness, and always with the goal of restoration.
You are also right about Paul and Barnabas. In Acts 15:36-40 we read that Paul and Barnabas had planned to revisit the brothers in every city where they had proclaimed the word. Barnabas wanted to take John Mark with them, but Paul insisted they should not take him because he had withdrawn from them in Pamphylia and had not gone with them to the work. Then the contention grew so sharp that they separated from each other. Barnabas took Mark with him, and sailed away to Cyprus, but Paul chose Silas, and went out, being commended by the brothers to the grace of God, Acts 15:39-40. This was a real disagreement between two godly apostles, and it led to a separation.
We also see this in Galatians 2:11 where Paul opposed Peter to his face because he stood condemned for withdrawing from eating with Gentile believers out of fear. So even apostles needed correction.
What we learn is that disagreement itself is not necessarily sin, but how we handle it matters. The separation of Paul and Barnabas was not the ideal of unity, yet God in His sovereignty used it to multiply the missionary effort into two teams, and later we see reconciliation. In 2 Timothy 4:11 Paul later says to bring Mark, for he is useful to me for service, showing that the relationship was restored and Mark was proven faithful. Iron sharpens iron, so a man sharpens the countenance of his friend, Proverbs 27:17, but sharpening can involve sparks. We are to contend for truth without holding onto bitterness, and to forgive and be reconciled as the Lord forgave us.
We also noticed your request did not mention the name of Jesus Christ. We want to gently remind you that it is only through Jesus that we have access to God and it is only by His name that we are saved. Jesus said in John 14:6 I am the way, the truth, and the life. No one comes to the Father, except through me. And in Acts 4:12 we are told there is salvation in no one else, for there is no other name under heaven that is given among men, by which we must be saved. We encourage you to place your trust fully in Jesus Christ, believing that He died for your sins and rose again, and to come to God the Father through Him in prayer.
We pray together for you now.
Father, we come before You in the name of Jesus Christ, Your Son, and we thank You for Your Word which is profitable for teaching, for reproof, for correction, and for instruction in righteousness, 2 Timothy 3:16. We ask that You give us wisdom to walk in love and unity while also having courage to exhort and correct one another in gentleness and truth. Help us to speak the truth in love and to bear with one another in patience and forgiveness. Where there has been sharp contention, heal and restore as You did with Paul, Barnabas, and Mark, and use even our disagreements for the furtherance of Your Gospel. Teach us to not let the sun go down on our anger and to be eager to keep the unity of the Spirit in the bond of peace. We ask that You draw the one who asked this question closer to Jesus, that they would know Him as Lord and Savior and come boldly to Your throne of grace through Him. Guide us all to exhort one another daily so that none of us would be hardened by the deceitfulness of sin. In Jesus name we pray, Amen.