Aims. We develop a new theoretical framework to generate Besançon Galaxy Model Fast Approximate Simulations (BGM FASt) to address fundamental questions of the Galactic structure and evolution performing multi-parameter inference. As a first application of our strategy we simultaneously infer the initial-mass function (IMF), the star formation history and the stellar mass density in the solar neighbourhood. Methods. The BGM FASt strategy is based on a reweighing scheme, that uses a specific pre-sampled simulation, and on the assumption that the distribution function of the generated stars in the Galaxy can be described by an analytical expression. To evaluate the performance of our strategy we execute a set of validation tests. Finally, we use BGM FASt together with an approximate Bayesian computation algorithm to obtain the posterior probability distribution function of the inferred parameters, by automatically comparing synthetic versus Tycho-2 colour-magnitude diagrams. Results. The validation tests show a very good agreement between equivalent simulations performed with BGM FASt and the standard BGM code, with BGM FASt being ∼10⁴ times faster. From the analysis of the Tycho-2 data we obtain a thin-disc star formation history decreasing in time and a present rate of 1.2 ± 0.2 M _⊙ yr⁻¹. The resulting total stellar volume mass density in the solar neighbourhood is 0.051_−0.005^+0.002 M_⊙ pc⁻³ and the local dark matter density is 0.012 ± 0.001 M _⊙ pc⁻³. For the composite IMF, we obtain a slope of α₂ = 2.1_−0.3^+0.1 in the mass range between 0.5 M_⊙ and 1.53 M_⊙. The results of the slope at the high-mass range are trustable up to 4 M_⊙ and highly dependent on the choice of extinction map (obtaining α₃ = 2.9_−0.2^+0.2 and α₃ = 3.7_−0.2^+0.2, respectively, for two different extinction maps). Systematic uncertainties coming from model assumptions are not included. Conclusions. The good performance of BGM FASt demonstrates that it is a very valuable tool to perform multi-parameter inference using Gaia data releases.