Was Paul's command to serve one another in love meant for believers' treatment of unbelievers?
No. Paul addressed Galatian believers in various congregations, instructing them to serve one another in love among themselves, believer with believer. The context indicates he was not necessarily commanding how they must work among the unbelieving world, but how they were to treat fellow believers as bond servants through love.
Les explains that Paul is writing to the Galatian believers who are in various congregations. Therefore, Galatians 5:13—"use not liberty for an occasion to the flesh, but by love serve one another"—is not necessarily talking about how they must work among the unbelieving world, but among themselves, believers with believers. They were to be like servants through love one to another. This fits the surrounding warning not to use Christian liberty as an occasion to the flesh, but instead to serve fellow believers in love.
For, brethren, ye have been called unto liberty; only use not liberty for an occasion to the flesh, but by love serve one another.