High fidelity GHZ states among remote nodes is a precious commodity which can allow for non-local stabilizer measurements and thus pave the way for a modular fault-tolerant quantum computer. To this end, we extend the high fidelity intracavity gate introduced by Borregaard et al. (2015) to distributed paradigm, consisting of SnV-inspired atomic states in cavities connected by fibers. The adiabatic dynamics of this system can be solved efficiently using the effective operator formalism of Reiter and Sørensen (2012). We develop a Python framework that enables the analytical calculation of these effective dynamics reliably and swiftly, while being versatile and easily modifiable. The possibilities of this framework are showcased by obtaining results for a symmetric distributed setup and verifying its scalability. We present the ways that it can be optimized while taking into consideration experimentally inspired constraints, and proceed to optimize it for GHZ generation in color centers. These optimized gates are compared against an emission based protocol using the GHZ creation simulations of the Modicum protocol. As a byproduct of our investigation, we identify a specific set of Hamiltonians which, under certain conditions, can generate GHZ states with a single multi-qubit entangling gate.