Inspired by biological processes, neuromorphic computing leverages spiking neural networks (SNNs) to perform inference tasks, offering significant efficiency gains for workloads involving sequential data. Recent advances in hardware and software have shown that embedding a small payload within each spike exchanged between spiking neurons can enhance inference accuracy without increasing energy consumption. To scale neuromorphic computing to larger workloads, split computing - where an SNN is partitioned across two devices - is a promising solution. In such architectures, the device hosting the initial layers must transmit information about the spikes generated by its output neurons to the second device. This establishes a trade-off between the benefits of multi-level spikes, which carry additional payload information, and the communication resources required for transmitting extra bits between devices. This paper presents the first comprehensive study of a neuromorphic wireless split computing architecture that employs multi-level SNNs. We propose digital and analog modulation schemes for an orthogonal frequency division multiplexing (OFDM) radio interface to enable efficient communication. Simulation and experimental results using software-defined radios reveal performance improvements achieved by multi-level SNN models and provide insights into the optimal payload size as a function of the connection quality between the transmitter and receiver.