Goodness-of-fit tests for the innovation distribution in GARCH models based on measuring deviations between the empirical characteristic function of the residuals and the characteristic function under the null hypothesis have been proposed in the literature. The asymptotic distributions of these test statistics depend on unknown quantities, so their null distributions are usually estimated through parametric bootstrap (PB). Although easy to implement, the PB can become very computationally expensive for large sample sizes, which is typically the case in applications of these models. This work proposes to approximate the null distribution through a weighted bootstrap. The procedure is studied both theoretically and numerically. Its asymptotic properties are similar to those of the PB, but, from a computational point of view, it is more efficient.