The nonlinear Schrödinger equation with a nonlocal operator plays an important role in quantum mechanics, and the time-fractional Schrödinger problems have been widely studied for the case of constant exponents. In this paper, we propose a linearized unconditionally convergent $L1$-Galerkin method to solve the variable-exponent fractional Schrödinger equations. The optimal error convergence of the fully discrete scheme is proved without any time-space step restriction condition, even when incorporating the influence of the nonlocal operator in the temporal direction. The proof relies critically on the Sobolev embedding theorem combined with the inverse inequality. The discrete fractional Grönwall inequality is also used to obtain the error estimates. Numerical experiments are given to verify our theoretical results.