We investigate the two-dimensional lattice U(1) gauge-Higgs model with a topological term, employing Lüscher's admissibility condition. The standard Monte Carlo simulation for this model is hindered not only by the complex action problem due to the topological term but also by the topological freezing problem originating from the admissibility condition. Resolving both obstacles simultaneously with the tensor renormalization group approach, we show the advantage of the admissibility condition in dealing with the topological term discretized with the so-called field-theoretical definition.