The incompressible stationary 2D Navier-Stokes equations are considered on an unbounded strip domain with a compact obstacle, and with the Poiseuille flow as a background flow near infinity. A computer-assisted existence and enclosure result for the velocity (in a suitable divergence-free Sobolev space) is presented, based on Newton-Kantorovich-like arguments. Starting from an approximate solution (computed with divergence-free finite elements), we determine a bound for its defect, and a norm bound for the inverse of the linearization at the approximate solution, which we obtain via eigenvalue bounds (supplemented by bounds on the essential spectrum) for some auxiliary self-adjoint eigenvalue problem. For computing these eigenvalue bounds, we use the Rayleigh-Ritz method, the Temple-Lehmann-Goerisch method, and a homotopy method for obtaining the needed spectral pre-information. In detail, three homotopies are performed; the first deforms the coefficients into piecewise constant ones, the second deforms the domain with obstacle into the full strip, and the third fades out the divergence condition.