In this work, it is proposed the direct and inverse analyses of the forced convection of an incompressible gas flow within rectangular channels in the range of the slip flow regime by taking into account the wall conjugation and the axial conduction effects. The Generalized Integral Transform Technique (GITT) combined with the single-domain reformulation strategy is employed in the direct problem solution of the three-dimensional steady forced convection formulation. A non-classical eigenvalue problem that automatically accounts for the longitudinal diffusion operator is here proposed. The Bayesian framework implemented with the maximum a posteriori objective function is used in the formulation of the inverse problem, whose main objective is to estimate the temperature jump coefficient, the velocity slip coefficient, and the Biot number, using only external temperature measurements, as obtained, for instance, with an infrared measurement system. A comprehensive numerical investigation of possible experimental setups is performed in order to verify the influence of the Biot number, wall thickness, and Knudsen number on the precision of the unknown parameters estimation.