  We investigate the phase structure of the (1+1)-dimensional U(1) gauge-Higgs
model with a $\theta$ term, where the U(1) gauge action is constructed with
L\"uscher's admissibility condition. Using the tensor renormalization group,
both the complex action problem and topological freezing problem in the
standard Monte Carlo simulation are avoided. We find the first-order phase
transition with sufficiently large Higgs mass at $\theta=\pi$, where the
$\mathbb{Z}_2$ charge conjugation symmetry is spontaneously broken. On the
other hand, the symmetry is restored with a sufficiently small mass. We
determine the critical endpoint as a function of the Higgs mass parameter and
show the critical behavior is in the two-dimensional Ising universality class.
