The well-known issue of reconstructing regulatory networks from gene expression measurements has been somewhat disrupted by the emergence and rapid development of single-cell data. Indeed, the traditional way of seeing a gene regulatory network as a deterministic system affected by small noise is being challenged by the highly stochastic, bursty nature of gene expression revealed at single-cell level. In previous work, we described a promising strategy in which network inference is seen as a calibration procedure for a mechanistic model driven by transcriptional bursting: this model inherently captures the typical variability of single-cell data without requiring ad hoc external noise, unlike ordinary or even stochastic differential equations often used in this context. The resulting algorithm, based on approximate resolution of the related master equation using a self-consistent field, was derived in detail but only applied as a proof of concept to simulated two-gene networks. Here we derive a simplified version of the algorithm and apply it, in more relevant situations, to both simulated and real single-cell RNA-Seq data. We point out three interesting features of this approach: it is computationally tractable with realistic numbers of cells and genes, it provides inferred networks with biological interpretability, and the underlying mechanistic model allows testable predictions to be made. A practical implementation of the inference procedure, together with an efficient stochastic simulation algorithm for the model, is available as a Python package.