Building on a work by Alm, we consider a model of weighted self-avoiding walks on a lattice and develop a method for computing upper bounds on the corresponding weighted connective constant, which we implement in a publicly available software package. The upper bounds are obtained as the dominant eigenvalues of certain matrices and provide detailed information about the convergence of the model's multivariate generating function. We discuss potential applications of our results to developing Peierls-type estimates for anisotropic contour models in statistical physics, generalizing a technique recently introduced by Fahrbach--Randall.