I think the answer is no and that actually the "finite quotient" commutator length (defined by your formula) is bounded on $[F_2,F_2]$.

Indeed by Nikolov-Segal, in the profinite completion $P$ of $F_2$, the derived subgroup $[P,P]$ is closed; since the set $C$ of commutators is compact, it follows by a Baire argument that $[P,P]$ is boundedly generated by $C$ (first get by Baire that some open neighborhood of the identity of $[P,P]$ has bounded commutator length in $P$ and then use compactness). Observe on the other hand that $[F_2,F_2]=[P,P]\cap F_2$, which is essentially trivial since the abelian group $F_2/[F_2,F_2]$ is residually finite. So in any finite quotient of $F_2$, the commutator length of $w\in [F_2,F_2]$ is bounded by the universal number that arises as upper bound of the commutator length of $[P,P]$ in $P$.

On the other hand, the commutator length of $F_2=\langle x,y\rangle$ is unbounded on $[F_2,F_2]$, as the commutator length of $[x,y]^n$ grows linearly (I think it's due to Bavard).