The main focus of this Small Compute Call 2024 Proposal is research into improving Matrix Product State Methods for quantum many-body physics. Here, we numerically study so called Spin-Holstein model (Knörzer et al., 2022) as an example of a relevant and challenging quantum many-body system. In particular, we aim to improve upon existing tensor network state (TNS) methods by applying a recently proposed integration of Gaussian state methods into TNS. (Nüsseler et al. 2021) We will reproduce the proposed approach, adapt it to Spin-Holstein models, and aim to develop further improvements of the method. The computational resources of this project will allow us to benchmark established and novel methods consistently against each other. In addition to this, the resources granted are used towards research in improved numerical methods for the study of Causal Fermion Systems (building on arXiv:2201.06382), hardware-optimised design of star-to-chain transformations for analogue quantum simulation or numerical simulations related to adiabatic quantum computing.