I am a project scientist and just saw that your question had no answer...
The climate models you are running are based on physical principles, or in other words, on equations (for example the conservation of mass and momentum). These equations basically calculate the values of temperature, precipitation, wind, etc, for the next time step, and then for the next one, and so on. There is no bayesian or frequentist approaches at this stage. We are running physically-based climate models, which are not the same as statistical models used to project future global mean temperature for example. Both are quite different techniques, and used to answer different types of questions.
Of course, with the outputs that you calculate, we perform statistical analyses to evaluate them. Depending again on the question we want to answer, we use frequentist or bayesian approaches. But often we simply start by calculating the mean and standard deviation of all ensemble members!