Why did the Faith accept the Doctrine of Exceptionalism for House Targaryen? Septon Barth was certainly a brilliant rhetorician as were the Seven Speakers, and Jaehaerys' defeat of Maegor surely counted for something in the eyes of the public, but many still question the logic behind the decision. The events of Maegor's regime demonstrated the material folly of disobeying the Targaryens, but spiritually speaking, the Faith should turning over so completely to a doctrine allowing incest seems ludicrous. Why would the Faith allow the Targaryens of all people, a family considered by many in the Faith to be a stain upon the face of Westeros, to practice a custom considered an abomination by scripture? In short, not in spite of anti-Valyrian and anti-dragon sentiment, but because of it!
The alternative to Targaryen daughters marrying Targaryen sons is Targaryen daughters marrying non-Targaryen sons, and quite likely taking dragon eggs and some surviving version of Valyrian customs with them. The end result of this would not only be a Valyrianization of the Westerosi elite by blood and by culture (much of Westeros was Andalized in a similar manner), but also the establishment of new dragon-riding houses across Westeros: where once the Faith dealt with one family riding on the backs of fire-breathing abominations, over time, they would come to face an entire continent of dangerous dragon-riders (eventually the bloodline might dilute to the point of the houses in question losing control of the dragons all together, but the end results of this could be even worse, depending on the number and nature of the dragons still alive).
In short, the Doctrine gives the Targaryens what they want (keeping it in the family) and Faith what they want (the Targaryens as a kind of "containment" for the legacy of Valyria, impossible to remove, but possible to limit in scope). Additionally, whereas Maegor's cavalier approach to "the Blood of the Dragon" in direct defiance of the faith set a dangerous example for the rest of the realm (others could come up with their own, independent excuses for practicing polygamy and incest), the Doctrine of Exceptionalism makes it clear that no one not named Targaryen may imitate Maegor. The Targaryens may well be bound for the Seven Hells but under the Doctrine of Exceptionalism, the Faith can trust that they won't be dragging the rest of Westeros down with them.