High dimensional Gaussian Process classifier: Edward or Stan?

You should use neither. There are many software dedicated to GPs, such as GPy, GPstuff, and GPflow. I recommend one of these as they have a suite of kernels and model-specific algorithms to handle common GP models including GP classification. (For example, I recommend GPy due to its maturity and abundant documentation.)

I recommend a probabilistic programming framework over GP-specific software only if you intend to scaffold GPs into more complex models, or incorporate GPs into a workflow that is more general than model-specific software.