How would it be betrayal, if the paper was signed, and you swore allegiance...but did not believe it yourself? Wouldn't God know what was in your heart?
IT would cause scandal to those who followed him as a bishop. It could lead them to disbelief, and as a father to his congregation he would have misled them. Christ said it would be better to die than to lead a child to sin, I believe this was both a literal and metaphorical statement.
In the end, it boils down to whether or not you believe truth is. What does it profit a man to gain the world but lose his soul?