I found this paper which uses a somewhat different approach from what I was thinking to do

DOI:10.1001/jama.282.11.1061

In particular in the materials and methods:

In summary, the dependent variable of the model was the logarithm DOR. Explaining variables were 2 parameters for each meta-analysis (the common DOR and the threshold parameter) and 9 covariates to examine the effect of the different study characteristics, 1 for each feature. All study characteristics were evaluated simultaneously in a multivariate model.

A weighted linear regression analysis was used, with weights proportional to the reciprocal of the variance of the log DOR. This weighted linear regression assumes fixed effects. […] The model was fitted using maximum likelihood estimation

I was wondering if instead of using the MLE approach one could fit a linear weighted model and reporting the coefficients of the covariate included as a study characteristic?

example log(DOR) = threshold_param + summary_DOR + study_characteristic