As a result of the ubiquitous network applications and services, exacerbated by the overarching digital revolution the need and demand for efficient and dependable connectivity solutions have surged to unprecedented levels. Quality of Service (QoS)-based routing has emerged as a critical solution, enabling service differentiation, efficient resource allocation, and improved network performance. In this study, we introduce a novel Genetic Algorithm-powered QoS-aware Cross-Network Traffic Engineering framework, GATE-BC, at the confluence of Software Defined Networking (SDN) and Blockchain (BC) technologies. GATE-BC orchestrates end-to-end (E2E) QoS traffic, providing resource-efficient, reliable, and latency-tolerant delivery of intelligent network services in BC-enabled SDNs. Leveraging BC features such as decentralization, transparency, and immutability, GATE-BC eliminates the need for centralized entities in QoS-supported cross-network routing models. We compare GATE-BC framework with three other traffic management and engineering approaches: QoSChain (QC), Hierarchical Routing Approach (HRA), and Distributed Routing Approach (DRA). The extensive simulations reveal that GATE-BC outperforms the other routing strategies in terms of Path Setup Time (PST), Network Message Overhead (NMO), Request Acceptance Ratio (RAR), Network Bandwidth Consumption (NBC), Average Path Length (APL), and Average Network Length (ANL) metrics under various network topologies. Furthermore, GATE-BC employs three different feasible path selection strategies based on bandwidth (GATE-BC_BW), delay (GATE-BC_D), and reliability (GATE-BC_R) QoS parameters to satisfy the service levels requested.