As other people in his court, with whom reciprocal vows of fealty have been made, heard his promise, he now has to keep his promise our of fear of them proactively removing him (or otherwise punishing him, such as by breaking their own vows of fealty) for breaking his promise.
Today one can lie with much more impunity, at best tying things up in court for years until the person you lied to gives up, at worst being ordered by the court to make good on the promise and pay some legal costs (maybe).
As your synopsis shows, it's not prestige that forced integrity, it's the fact that others heard, and presumably would enforce such vows, even against a king, that forced integrity, even from a king.
I don't think a direct threat of enforcement is the problem. A king who was known to have gone back on his word would be an illegitimate ruler and he would be unable to function as king within his court and his kingdom. He would end up getting deposed, but that wouldn't be part of an effort to enforce the king's promise, it would just be what happens to kings who aren't fit to rule.
The direct consequence of the king breaking his vow is "only" that the king loses his prestige. But the indirect consequences are severe; "king" is a position from which it is not safe to be removed.
