Edge nodes in the Industrial Internet of Things (IIoT) often face a fundamental trade-off between limited computational resources and stringent real-time inference requirements. Moreover, sensitive data they generated are exposed to significant privacy and security threats during transmission and computation. To address these challenges, this paper proposes a lightweight additive homomorphic edge neural network framework called LAHENet. This framework achieves millisecond-level inference latency in real-world industrial environments through a combination of a dual-metric feature selection strategy, an efficient additive homomorphic signcryption protocol, and a lightweight linear computation layer with adaptive layer collapsing. It ensures end-to-end confidentiality, unforgeability, forward security, and verifiable computation correctness. Experimental results show that LAHENet maintains a constant communication overhead at a few kilobytes per inference while preserving high model accuracy. It significantly enhances inference efficiency and reduces bandwidth consumption in edge environments, offering a practical private inference solution for large-scale IIoT deployments.