Using a Wald likelihood ratio test to select model parameters will invalidate any p-values and confidence intervals, as well as biasing estimates - unless you adjusted for that somehow. Thus, the reviewer is right to highlight this topic, if you had not done some appropriate there - especially so, if the model building is an important part of the paper. Just reporting estimate, CI and p-value after model selection as if not model selection had occured is wrong and misleading.
I am not sure that using a Bonferroni-Holm correction is an approach for dealing with that. Perhaps it is possible to use it for that purpose at least to correct the p-values, but it may be extremely conservative.
If you then afterwards are performing multiple comparisons across multiple levels of a random effect, then there's some people that would argue that no correction is needed (see e.g. Gelman et al.'s "Why we (usually) don't have to worry about multiple comparisons" article) - but others (perhaps including your reviewer) may disagree. If you are comparing multiple levels of some fixed effects factor, then it is clearer that there is a multiple comparison issue (although some might still argue against the need for type I error correction if this is rather exploratory and you do not want to use a hypothesis testing approach claiming "statistical significance").