If I understand the question correctly, you are concerned that there might be hidden relations between the elements of $T$. Moreover, since the images of $T$ are linearly independent in the abelianization, these would have to be commutator relations.

This kind of this can't happen: if there were a nontrivial relation involving elements of $T$, then said relation would involve only finitely many elements of $T$, so it would be a relation on the free subgroup generated by that finite set of elements. Essentially, condition (2) guarantees that the elements of any finite subset of $T$ are algebraically independent, and therefore all of the elements of $T$ are algebraically independent.

If I understand the question correctly, you are concerned that there might be hidden relations between the elements of $T$. Moreover, since the images of $T$ are linearly independent in the abelianization, these would have to be commutator relations.

This kind of thing can't happen: if there were a nontrivial relation involving elements of $T$, then said relation would involve only finitely many elements of $T$, so it would be a relation on the free subgroup generated by that finite set of elements. Essentially, condition (2) guarantees that the elements of any finite subset of $T$ are algebraically independent, and therefore all of the elements of $T$ are algebraically independent.