I think we are going to see one of the dragons breathe fire on Jon (on purpose with the intent to kill for something the latter does that angers The Mad Queen). Presumably he would survive. It's the only way forward that I can see since a Maester's book and Bran's visions are not going to persuade anyone.
My only worry is that the dragons probably already know Jon is a Targaryen based on smell or whatever other fantasy magic trope the show's writers want to use.