Blockchain offers a decentralized trust framework for the Internet of Things (IoT), yet deploying consensus in spectrum-congested and dynamic wireless edge IoT networks faces fundamental obstacles: traditional BFT protocols are spectrum-ignorant, leading to inefficient resource utilization and fragile progress under time-varying interference. This paper presents \textit{Wireless Streamlet}, a spectrum-aware and cognitive consensus protocol tailored for wireless edge IoT. Building on Streamlet's streamlined structure, we introduce a \textit{Channel-Aware Leader Election (CALE)} mechanism. CALE serves as a verifiable cross-layer cognitive engine that leverages receiver-measured channel state information (CSI) piggybacked in signed votes to derive Byzantine-robust connectivity scores from notarization certificates, and deterministically selects a unique weighted leader per epoch from finalized history, thereby improving proposal dissemination reliability under deep fading. Complementing this cognitive adaptation, Wireless Streamlet exploits the single-hop broadcast medium and a deterministic TDMA voting schedule to achieve linear per-epoch on-air transmissions (slot complexity), ensuring deterministic spectral access. To address the communication-storage trade-off, we further propose a coded dual-chain architecture that decouples header-only consensus (State Chain) from payload data (Data Chain). By employing erasure coding and on-chain integrity commitments, the system minimizes redundant spectrum usage for data retrieval while ensuring availability. Experiments show that Wireless Streamlet achieves higher throughput and lower confirmation latency than representative baselines in lossy environments, while substantially reducing per-node storage, demonstrating the efficacy of integrating cognitive sensing into consensus logic.