The experimental discovery of single-pulse ultrafast magnetization switching in ferrimagnetic alloys, such as GdFeCo and MnRuGa, opened the door to a promising route toward faster and more energy efficient data storage. A recent semi-phenomenological theory has proposed that a fast, laser-induced demagnetization below a threshold value puts the system into a dynamical regime where angular momentum transfer between sublattices dominates. Notably, this threshold scales inversely proportional to the number of exchange-coupled nearest neighbors considered in the model, which in the simplest case is directly linked to the underlying lattice structure. In this work, we study the role of the lattice structure on the laser-induced ultrafast magnetization switching in ferrimagnets by complementing the phenomenological theory with atomistic spin dynamics computer simulations. We consider a spin model of the ferrimagnetic GdFeCo alloy with increasing number of exchange-coupled neighbors. Within this model, we demonstrate that the laser-induced magnetization dynamics and switching depend on the lattice structure. Furthermore, we determine that the critical laser energy for switching reduces for decreasing number of exchange-coupled neighbors.