A linear and second order numerical scheme is proposed and analyzed for a droplet thin film coarsening model, with a singular Leonard-Jones energy potential involved. This numerical scheme with unconditional energy stability is based on the backward differentiation formular (BDF) method in time derivation combining with the stabilized scalar auxiliary variable approach in time and the Fourier spectral method in space. A second order accurate artificial regularization term, in the form of $S∆(\phi^{n+1}− 2\phi^n + \phi^{n−1}),$ is added in the numerical scheme to make better stability of the numerical scheme. Moreover, we present the detail proof for unconditional energy stability property of the numerical scheme, without any restriction for the time step size. In addition, an $\mathcal{O}(∆t^2+h^m)$ rate convergence estimate in the $ℓ^∞(0, T ; ℓ^2)$ norm are derived in details with the help of a priori assumption for the error at the previous time step. Some numerical simulation results are presented to demonstrate the efficiency and accuracy of the second-order numerical scheme.