To what does "God's coworker" refer?

The second title that Paul uses for Timothy is God’s coworker (Greek: συνεργὸν τοῦ θεοῦ/sunergon tou theou). There is some uncertainty over what Paul actually wrote here. Many manuscripts have the words God's servant (Greek: διάκονον τοῦ θεοῦ/diakonon tou theou) and some have simply coworker (Greek: συνεργόν/sunergon) rather than God’s coworker. It is not easy to choose between these options, but the most likely explanation for the different readings is that Paul wrote God's coworker and that later scribes felt uncomfortable with such a designation for Timothy; as a result, they altered the text to read either God's servant or simply coworker.¹

Fee² and Green³ who wish to avoid the idea that Timothy was God’s coworker, try to connect the possessive pronoun our to both the terms brother and fellow worker. Green appeals to Philippians 2:25 in support of this interpretation. However, the grammar of 1 Thessalonians 3:2 really does not allow this. The Greek text has two balanced phrases our brother (Greek: τὸν ἀδελφὸν ἡμῶν/ton adelphon hēmōn) and God's fellow worker (Greek: συνεργὸν τοῦ θεοῦ/sunergon tou theou) separated by and (Greek: καὶ/kai), where the most natural interpretation is that each possessive (our, God's) belongs to the noun in its own phrase. If the possessive pronoun our (Greek: ἡμῶν/hēmōn) was meant to apply to both nouns, the most natural way of expressing this would have been to write the two nouns together followed by the pronoun: τὸν ἀδελφὸν καὶ σύνεργὸν ἡμῶν/ton adelphon kai sunergon hēmōn. This is the form that is used in Philippians 2:25. Furthermore, to take our with both nouns leaves the second possessive (God's) either unattached to any noun or clumsily attached to a noun that is already qualified by another possessive. If Paul had meant to say our brother and coworker—belonging to God (as Fee contends) he could not have done so in a more confusing way!