This paper introduces SpikeFit, a novel training method for Spiking Neural Networks (SNNs) that enables efficient inference on neuromorphic hardware, considering all its stringent requirements: the number of neurons and synapses that can fit on a single device, and lower bit-width representations (e.g., 4-bit, 8-bit). Unlike conventional compressing approaches that address only a subset of these requirements (limited numerical precision and limited number of neurons in the network), SpikeFit treats the allowed weights' discrete values themselves as learnable parameters co-optimized with the model, allowing for optimal Clusterization-Aware Training (CAT) of the model's weights at low precision (2-, 4-, or 8-bit) which results in higher network compression efficiency, as well as limiting the number of unique synaptic connections to a value required by neuromorphic processor. This joint optimization allows SpikeFit to find a discrete weight set aligned with hardware constraints, enabling the most complete deployment across a broader range of neuromorphic processors than existing methods of SNN compression support. Moreover, SpikeFit introduces a new hardware-friendly Fisher Spike Contribution (FSC) pruning method showing the state-of-the-art performance. We demonstrate that for spiking neural networks constrained to only four unique synaptic weight values (M = 4), our SpikeFit method not only outperforms state-of-the-art SNNs compression methods and conventional baselines combining extreme quantization schemes and clustering algorithms, but also meets a wider range of neuromorphic hardware requirements and provides the lowest energy use in experiments.