You raise important points. Meta-analysis should not be performed at any price. If the studies are too different to be pooled in a meta-analytic model, this is per se a very important information (with ending conclusion of the study that several points should be address in the future to be able to conduct meta-analysis). Conducting meta-analysis there would not have any sense (the pooled estimate means nothing when you mix pear and oranges....). We have the same problem in different topics in animal health where either the methodology differed from a study to another or when there are too few studies to be able to put them in a model.

If the protocol included specific quality criteria and comparability to perform the meta-analysis it is OK to only report these studies also focusing on risk of publication bias. depending on the specific topic and if study have prespecified criteria to be included in the meta-analysis a meta-regression using study quality as a covariate could also be conducted to look for the impact of study quality on the outcome of interest.

Meta-analysis is a mathematical method of pooling the results of several or more studies; a meta-analysis may be based on a systematic review, but this is not always the case. A systematic review is a multistage process aimed at the identification of all reliable evidence regarding a specific clinical problem. Systematic reviews make it possible to objectively address particular issues according to the current state of clinical knowledge and therefore constitute a reliable basis for clinical decision-making. An appropriate systematic review should include: 1) a defined clinical question, 2) pre-specified inclusion and exclusion criteria, 3) complex search for medical evidence sources according to a search strategy, 4) critical evaluation of reliability of identified clinical trials, 5) qualitative or quantitative data synthesis and 6) evidence based conclusions. These simple criteria, formulated by Cook et al. more than 10 years ago, allow to differentiate between a reliable systematic review and a "quasi-systematic" one, as well as between a reliable meta-analysis based on a systematic review and a potentially misleading meta-analysis without a systematic review.

Review articles, systematic reviews and meta-analyses: which can be trusted? - PubMed (nih.gov)

It seems okay to include articles in the systematic review (only systematic review). However, if you are going to do a meta-analysis, as it is known, the included studies must first meet some criteria. In this respect, it is an expected fact that the majority of the studies are of high quality. The included studies are also expected to be of high quality, as meta-analyses demonstrate a high level of evidence. This is an important issue. Meta-analysis with studies of low quality and not meeting the inclusion criteria is not appropriate and may lead to publication bias.

Article Meta-Analytic Methodology for Basic Research: A Practical Guide

https://www.ccace.ed.ac.uk/research/software-resources/systematic-reviews-and-meta-analyses

Article Quality Control in Systematic Reviews and Meta-analyses

Article Methodological Guidance Paper: High-Quality Meta-Analysis in...

If the moderate quality studies tend to be of moderate quality for different reasons when compared to each other (i.e. they are not of moderate quality because all or most of them share the same type of methodological flaw), then meta-analysis may be justified. Meta-analysis assumes that the flaws differ from one study to another and are not the same flaws (e.g., non-random sampling) that occur repetitively and consistently across the moderate quality studies. When the flaws are different, they are likely to contribute different levels and types of bias that will be factored out of the estimated pooled effect across studies. In this situation, the pooled effect is not biased by these different flaws with heterogeneous effects across studies, in contrast to when the same type of flaws recur across most or all of the studies and their similarity and consistency means they cannot be factored out of the unbiased pooled effect.

In your case, I would do a meta-analysis of all studies together, then do a subgroup analysis of high vs moderate vs low quality studies. Then, you can discuss the differences and limitations in the discussion, and let the reader decide.