I can look a bit more into this on Monday.
Generally, you can use weights in Stan as you can do it in when computing maximum likelihood estimates. (Though not everyone agrees one should)
The reason I’m am hesitant in my answer is, that I am unsure about what the sigma is in the inverse variance weights. I assume it’s the sigma of the effect sizes, and I mm addition one estimates an error variance, but I can’t tell without access to the paper. (I am also not sure what tau is).
As an aside, you can use brms to estimate meta analysis models. See for example here: https://vuorre.netlify.com/post/2017/01/19/better-forest-plots-from-meta-analytic-models-estimated-with-brms/