Here's a list of what you had to do until this point to get him to do it:
- Treat him with respect as a prisoner of war
- Give him aid and information against devils in his lands
- Heal his daughter of a seemingly incurable illness with no immediate request for a reward
- Protect his daughter against the dangers of magic
- Actively work to defend the Seven Kingdoms as their king even though you had none of the privileges of kingship
Stannis is stubborn, enormously, impossibly stubborn, but he is not insane, he can see the world under his nose and the fact of that matter is Viserys is a more worthy king in his eyes than Robert, just as Robert was more worthy than Aerys during the rebellion. I bears remebering that Stannis has broken oaths of fealty before
in extremis.
That said all this disbelief makes me wonder if I should have written this as a Stannis interlude.