Not an answer but this is how I would proceed:
Background knowledge: Given category $C$ and a (pseudo-) functor $F:C^{op}\to \mathrm{Cat}$ one can construct a category $$\int F$$ (also denoted $\int_C F$) called the Grothendieck Construction that comes with a canonical arrow $\pi_f:\int F\to C$ that happens to be a fibration. Inversely: Given a fibration, $\pi: E\to C$ we can construct a (pseudo-) functor $F_\pi:C^{op}\to \mathrm{Cat}$. These two constructions are inverse in a suitable way. Similarly for covariant functors $F:C\to\mathrm{Cat}$.
Now: To check if your functor $\mathrm{LiePs}\to\mathrm{Alg}$ is a fibration I would try to construct $\mathrm{LiePs}$ as the grothendieck construction of a functor $$Ps:\mathrm{Alg}^{op}\to\mathrm{Cat}$$ (or maybe $Ps:\mathrm{Alg}\to\mathrm{Cat}$).
The object part of this functor should be
$$Ps: A \mapsto \mathrm{Lie}^*_{/\mathrm{Der}(A)}$$ where $\mathrm{Lie}^*_{/\mathrm{Der}(A)}$ is a suitable subcategory of the comma category $\mathrm{Lie}_{/\mathrm{Der}(A)}$. A nice feature of the Grothendieck construction is the following: If your functor actually is a fibration, you should then be able to extend this assignment to a functor in a more or less obvious way.
The next step is to check if the resulting category $\int_{\mathrm{Lie}} Ps$ is the "same" as $\mathrm{LiePs}$. From the construction of $\int_{\mathrm{Lie}} Ps$ there should already by suitable candidates for the equivalence functors.