My answer to your question here included a proof of this. Indeed, such a group has a non-abelian 2-nilpotent quotient. In the discrete case, it even shows that there is a non-abelian torsion-free 2-nilpotent quotient. In the profinite case, it shows that for all $p$ (although I gave a full proof only for odd $p$) there is a non-abelian 2-nilpotent $p$-group quotient. It still works for a (profinite or discrete) group on $n$ generators with any family of relators such that at most $n-2$ of the relator do not belong to $[F,[F,F]]$, where $F$ is free (discrete or profinite) on $n$ generators.

Of course this gives no info about largeness.

My answer to your question here included a proof of this. Indeed, such a group has a non-abelian 2-nilpotent quotient. In the discrete case, it even shows that there is a non-abelian torsion-free 2-nilpotent quotient. In the profinite case, it shows that for all $p$ (although I gave a full proof only for odd $p$) there is a non-abelian 2-nilpotent $p$-group quotient. It still works for a (profinite or discrete) group on $n$ generators with any family of relators such that at most $n-2$ of the relator do not belong to $[F,[F,F]]$, where $F$ is free (discrete or profinite) on $n$ generators.

Of course this gives no info about largeness.