<p>In this work, we study a (3+1)-dimensional integrable system that unifies the Korteweg-de Vries (KdV) and Calogero-Bogoyavlenskii-Schiff (CBS) equations, with the aim of systematically constructing interaction solutions and revealing their complex dynamics. To this end, we propose a bilinear Bäcklund neural network method (BBNNM) that innovatively encodes Bäcklund transformations into a neural-network parametrization of the tau functions. The resulting algebraic constraints are solved analytically, enabling the direct generation of new closed-form solutions from known seeds without any iterative training. Crucially, we report the first-ever derivation of two types of bilinear Bäcklund transformations for the (3+1)-dimensional KdV-CBS equation. This framework breaks the constraints of the conventional exponential-function ansatz, allowing for the unified construction of substantially richer families of Bäcklund-associated interaction solutions. By applying the BBNNM to the KdV-CBS equation, we obtain several classes of explicit interaction solutions, including breather waves, breather-lump interactions, bright-dark soliton interactions, and complexiton solutions. Their spatiotemporal evolution and interaction patterns are illustrated through time-dependent three-dimensional plots and density maps. Furthermore, by coupling the constructed wave profiles with a forced Duffing oscillator, we analyze the emergence of chaotic wave behaviour and characterize the corresponding phase-space dynamics. The results not only deepen the understanding of nonlinear wave interactions and chaos in high-dimensional integrable systems but also provide a flexible symbolic-computation-based framework that can be extended to other nonlinear wave models and may offer insights for controlling chaotic dynamics in related physical settings.</p>