As far as I know, the state of the art gives some improvements to those bounds: Given a nontrivial element $w \in F_2$ of word length $\ell$, using a result of Buskin (Economical separability in free groups, Sib. Math. J., 50 (2009), 603-608) there exists a subgroup, $H$, of index $\ell/2+2$ that does not contain $w$. By looking at the action of $F_2$ on $F_2/ H$ we get a representation of $F_2$ into $S_{\ell/2+2}$, that does not kill $w$. Therefore, in order for $w$ to be trivial in any representation of $F_2$ into $S_n$ we must have that $n \leq \ell /2 + 2$, or $2(n-2) \leq \ell$.

There are also better upper bounds known (see, for instance, Asymptotic growth and least common multiples in groups (me and Ben McReynolds), Bulletin of the LMS (2011)).

Your question is equivalent to quantifying residual finiteness of free groups (the non-normal case), for which the precise answer is still unknown (the best known bounds are from the papers above).

As far as I know, the state of the art gives some improvements to those bounds, but they are not huge. For a better lower bound: given a nontrivial element $w \in F_2$ of word length $\ell$, using a result of Buskin (Economical separability in free groups, Sib. Math. J., 50 (2009), 603-608) there exists a subgroup, $H$, of index $\ell/2+2$ that does not contain $w$. By looking at the action of $F_2$ on $F_2/ H$ we get a representation of $F_2$ into $S_{\ell/2+2}$, that does not kill $w$. Therefore, in order for $w$ to be trivial in any representation of $F_2$ into $S_n$ we must have that $n \leq \ell /2 + 2$, or $2(n-2) \leq \ell$.

There are also better upper bounds known (see, for instance, Asymptotic growth and least common multiples in groups (me and Ben McReynolds), Bulletin of the LMS (2011)).

Your question is equivalent to quantifying residual finiteness of free groups (the non-normal case), for which the precise answer is still unknown (the best known bounds are from the papers above).