No, even if $\|S^2_n x-S^2x\|_{F_2}\to0$ for some $S^2\in\mathcal L(E_2,F_2)$ and all $x\in E_2$.
Indeed, let e.g. $E_1=F_1=E_2=F_2=\ell^2$, $S^1_n=I$, $S^1=I$, $S^2=0$, and $S^2_n x=(e^{-|k-n|}x_k)_{k=1}^\infty$ for $x=(x_k)_{k=1}^\infty\in\ell^2$. Then all the conditions hold, but the conclusion $\text{ker}S^2=\{0\}$ does not hold.