I think this is possible (over any base field $k$, not just $\mathbb{F}_p$). You can rephrase your question to ask: if an isomorphism $\phi: k[[x_1, \dots, x_d]]/I \to k[[y_1, \dots, y_d]]/J$ is specified, when can it be lifted to an isomorphism $k[[x_1, \dots, x_d]] \to k[y_1 , \dots, y_d]]$? Or even further, if you're given a surjection $$\pi: A = k[[x_1, \dots, x_d]] \to k[[y_1, \dots, y_d]]/J = B,$$ when does it come from an isomorphism $$\widetilde{\pi} : A \to C = k[[y_1, \dots, y_d]]?$$ Perhaps this follows from some "formal" properties, but in any case it's not a difficult exercise. You work with the map on tangent spaces
$$
\overline{\pi} : {\frak{m}}_A/{\frak{m}}_A^2 \to {\frak{m}}_B/ {\frak{m}}_B^2 \cong {\frak{m}}_C/ (J \cap {\frak{m}}_C + {\frak{m}}_C^2)
$$
which must be surjective. So you just have to extend it to an isomorphism of the $d$-dimensional vector spaces, by choosing an isomorphism of $K = \mathrm{ker}(\overline{\pi})$ with $(J \cap {\frak{m}}_C + {\frak{m}}_C^2)/ {\frak{m}}_C^2$, and then it will automatically induce an isomorphism $\widetilde{\pi}$.