We present a novel Bayesian method for the joint reconstruction of cosmological matter density fields, peculiar velocities and power-spectra in the quasi-nonlinear regime. We study its applicability to the Ly-alpha forest based on multiple quasar absorption spectra. Our approach to this problem includes a multiscale, nonlinear, two-step scheme since the statistics describing the matter distribution depends on scale, being strongly non-Gaussian on small scales (< 0.1 h^{-1} Mpc) and closely lognormal on scales >~10 h^{-1} Mpc. The first step consists on performing 1D highly resolved matter density reconstructions along the line-of-sight towards z~2-3 quasars based on an arbitrary non-Gaussian univariate model for matter statistics. The second step consists on Gibbs-sampling based on conditional PDFs. The matter density field is sampled in real space with Hamiltonian-sampling using the Poisson/Gamma-lognormal model, while redshift distortions are corrected with linear Lagrangian perturbation theory. The power-spectrum of the lognormal transformed variable which is Gaussian distributed (and thus close to the linear regime) can consistently be sampled with the inverse Gamma distribution function. We test our method through numerical N-body simulations with a computational volume large enough (> 1 h^{-3} Gpc^3) to show that the linear power-spectra are nicely recovered over scales larger than >~20 h^{-1} Mpc, i.e. the relevant range where features imprinted by the baryon-acoustics oscillations (BAOs) appear.