Recent detections of Highly Pathogenic Avian Influenza (HPAI) in dairy cattle have underscored the urgent need to understand within-herd transmission dynamics and potential cross-species spillover risks. This study develops a deterministic $SEI_{s}I_{a}RB$ compartmental model specifically designed to capture the complex interplay between susceptible cattle, multiple infectious states, and environmental reservoirs. By employing the next-generation matrix method, we analytically derive the basic reproduction number ($R_0$) and identify the critical thresholds governing disease extinction and persistence. Rigorous local and global stability analyses of the disease-free and endemic equilibria are provided to characterize long-term herd health outcomes and steady-state behaviour. Numerical simulations validated by comprehensive sensitivity analysis utilising Latin Hypercube Sampling (LHS) and Partial Rank Correlation Coefficients (PRCC)---reveal that the transmission rate from asymptomatic infectious cattle ($\beta_a$) is the most influential parameter driving the epidemic curve. Furthermore, our results indicate that the environmental pathogen load significantly sustains the outbreak duration. These findings demonstrate that traditional surveillance focusing solely on symptomatic animals may significantly underestimate the true scale of an outbreak. Consequently, we propose that effective HPAI mitigation strategies must integrate rigorous environmental sanitation protocols with enhanced diagnostic screening for asymptomatic carriers to prevent wide-scale agricultural disruptions and broader public health impacts.