Is it an appropriate way to set random seed as hyperparameter in model selection?

I am using Bayesian linear regression model y= beta*x +alpha with inference by Markov Chain Monte Carlo (MCMC). I use different random seed to start MCMC, I found that the mean of regression coefficients vector beta are close, relative error about 3%. However, if I want to calculate the probability P{y_new<0|x_new, beta, alpha} with out-of-sample data, the results can be 0.39, 0.49, 0.50, 0.55, etc varies with different random seeds.So is it appropriate to add the random seed as a hyperparameter to model selection process by cross validation? Or is the model just not suitable to do prediction?