How do I find out the full path of a page’s template?

Unfortunately there is no way to do this. In the end it is nothing more but a meta field. It can come from the current theme, and old theme that doesn’t even exist any more or a plugin. Since 4.4.0 it can even be anything – it doesn’t even need to be a filename.

Sorry to tell you, but you can’t have what you’re asking for.