Not a direct answer but this is how I would tackle this:

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**Background knowledge:** Given a category $C$ and a (weak-) functor $F:C^{op}\to \mathrm{Cat}$ one can construct a category $$\int F$$ (also denoted $\int_C F$) called the [Grothendieck Construction][1] 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 (weak-) functor $F_\pi:C^{op}\to \mathrm{Cat}$. These two constructions are inverse in a suitable way. Similarly for covariant functors $F:C\to\mathrm{Cat}$.

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**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 weak functor in a more or less obvious way: For every morphism $f:A\to B$ in $\mathrm{Alg}$ you would search for a functor $f^*:\mathrm{Ps}(B)\to \mathrm{Ps}(B)$ (Or maybe covariantly $f_*:\mathrm{Ps}(A)\to \mathrm{Ps}(B)$?).

The next step is to check if the resulting category $\int Ps$ is the "same" as $\mathrm{LiePs}$. From the construction of $\int Ps$ there should already by suitable candidates for the equivalence functors.


  [1]: https://en.wikipedia.org/wiki/Grothendieck_construction