So I skimmed the paper you referred to, as well as one of the authors’ blog posts about it, and I’ll say first that it will not be possible to implement their scheme in Stan in any kind of straightforward way.
Second, the entire idea of their approach is really only applicable for cases where you have un-explorable/un-mitigatable model misspecification, which is a scenario that I think rarely (if ever) holds. I may be failing to consider the range of possible real-world scenarios, but if you know that one aspect of your model is more likely than others to be misspecified, it very sketchy to me to simply attempt to limit/bias the means by which information from that part of the model affects other parts of the model and push on with inference from the latter. Best practice that seems to be the growing culture in the Stan-iverse at least is to explore misspecification when you encounter it and add structure as necessary.