I don’t have a great reference for this.

A place to start might be Judea Pearl’s essay “Why I’m only half-Bayesian” at https://ftp.cs.ucla.edu/pub/stat_ser/r284-reprint.pdf . If you look at his Twitter account at @yudapearl, you will also see numerous tweets where he refers to Bayes Theorem as a “trivial identity” and where he talks about Bayesian statistics as “spraying priors on everything”. See for example https://twitter.com/yudapearl/status/1143118757126000640 and his discussions with Frank Harrell.

Another good read may be Robins, Hernan and Wasserman’s letter to the editor at Biometrics, https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4667748/ . While that letter is not about graphical models, the propensity scores/marginal structural models are mathematically very closely related. The main argument in that letter (which was originally a blog post) has been discussed on Less Wrong before; I am trying to find the discussion, it may be this link https://www.lesswrong.com/posts/xdh5FPMYYGGX7PBKj/the-trouble-with-bayes-draft

From my perspective, as someone who is not well trained in Bayesian methods and does not pretend to understand the issue well, I just observe that methodological work on causal models very rarely uses Bayesian statistics, that I myself do not see an obvious way to integrate it, and that most of the smart people working on causal inference appear to be skeptical of such attempts

I look forward to reading it. To be honest: Knowing these authors, I’d be surprised if you have found an error that breaks their argument.

We are now discussing questions that are so far outside of my expertise that I do not have the ability to independently evaluate the arguments, so I am unlikely to contribute further to this particular subthread (i.e. to the discussion about whether there exists an obvious and superior Bayesian solution to the problem I am trying to solve).