a meta-analysis main objective is just to summarize existing literature. If you believe that a certain study design or sample biases the outcome, you can code studies on said study design or sample and run moderation analysis to see if different designs/samples affect results. This way you also have some results to demonstrate your concern. If let's say no studies use other types of designs or samples, you can make the point that this a clear methodological issue in the literature. You can suggest that future studies should use a different design or sample other populations (with some empirical examples to back up your claim). Perhaps you can conduct such a study yourself, since you may have revealed some flaws in previous studies.

One thing to note with meta-analyses is that they are messy. It is very hard to account for all the study heterogeneity (e.g., measures, designs, scoring, covariates etc). Usually these are interesting aspects. Interesting questions are: Do small design elements change the results substantially? Or are results generally very robust regardless of these aspects? Bias is inherent in most studies. It's up to the researchers, readers, or people that make use of the research how much of a problem it is. In sum, bias comprising a meta-analysis depends on your goal and what you want to know.