I'd be happy to be corrected but I think you would generate P-values the same way you check the accuracy of all climate models, you use the model to make predic

I'd be happy to be corrected but I think you would generate P-values the same way you check the accuracy of all climate models, you use the model to make predictions about the past and compare them to measured values. If model parameters are unknown, you should be doing a sensitivity analysis on these parameters anyway (you must have already estimated values for these to run your model in the first place). Once you have a measure of the uncertainty of your model and sensitivity to unknown parameters, you can combine this with probabilities of different warming scenarios (maybe from IPCC data), and determine the likelihood of the circulation stopping given current emissions/warming trends.