A coherent transformation between ultracold atoms and molecules, including between completely bosonic condensates, is enabled by the magneto-association of atoms at a Feshbach resonance, which results in a passage through a quantum critical point. In this study, we show that the presence of generic interactions between the constituent atoms and molecules can fundamentally alter the nature of the critical point, change the yield of the reaction, and modify the order of the consequent phase transition. Furthermore, we demonstrate that non-adiabatic excitations in a nonlinear quantum phase transition can be described by the sum of Kibble-Zurek scaling and a first-order correction, similarly to the result presented in Phys. Rev. Lett. 132, 241601 (2024).