Edge-cloud co-inference enables efficient deep neural network (DNN) deployment by splitting the architecture between an edge device and cloud server, crucial for resource-constraint edge devices. This approach requires balancing on-device computations and communication costs, often achieved through compressed intermediate feature transmission. Conventional DNN architectures require continuous data processing and floating-point activations, leading to considerable energy consumption and increased feature sizes, thus raising transmission costs. This challenge motivates exploring binary, event-driven activations using spiking neural networks (SNNs), known for their extreme energy efficiency. In this research, we propose SpikeBottleNet, a novel architecture for edge-cloud co-inference systems that integrates a spiking neuron model to significantly reduce energy consumption on edge devices. A key innovation of our study is an intermediate feature compression technique tailored for SNNs for efficient feature transmission. This technique leverages a split computing approach to strategically place encoder-decoder bottleneck units within complex deep architectures like ResNet and MobileNet. Experimental results demonstrate that SpikeBottleNet achieves up to 256x bit compression in the final convolutional layer of ResNet, with minimal accuracy loss (0.16%). Additionally, our approach enhances edge device energy efficiency by up to 144x compared to the baseline BottleNet, making it ideal for resource-limited edge devices.