We establish the Lifshitz singularity of the integrated density of states (IDS) for random Schrödinger operators \[ H^ω = φ(-\mathcal{L}) + V^ω \] on planar unbounded nested fractals with the Good Labeling Property. Here, $\mathcal{L}$ is the Laplacian on the fractal, $φ$ is an operator monotone function with mild regularity, and $V^ω$ is a Poissonian random potential with a sufficiently regular profile. The main novelty of our work lies in showing that the study of $V^ω$ can be effectively reduced to the analysis of certain alloy-type potential, where the sites are no longer lattice points as in the classical $\mathbb{Z}^d$ case, but fractal complexes. This observation enables us to apply an approach, new in the setting of Poissonian random fields, which allows us to treat a broad class of Bernstein functions $φ$. In particular, it covers the case $φ(λ)=(λ+m^{d_w/\vartheta})^{\vartheta/d_w}-m$, $\vartheta \in (0,d_w)$, $m>0$, corresponding to relativistic models, which were previously unattainable on fractals by known methods.