 Miyabi and Buddhism seem to be rather contradictory values. Miyabi deals primarily with outer appearance and the worldly goal of a prestigious name and stature in the royal court, while Buddhism deals with the next life and the spiritual goal of Nirvana. They seem polar opposites, but both appear to be present in the life styles of the Japanese court, at least enough so that they appear in literature of the time. But while Buddhism is a spiritual quest that deals with the future and things unseen, miyabi is a practical value for everyday life, and so seems to be more deeply integrated into the minds of the people at court who make a claim for both.
The appreciation for religion is an aware virtue, I think. "Yes it's beautiful but it won't last" is a relatively Buddhist approach to the world; this life won't last so don't love it too much; think about the next one. Aware, demanded as a social virtue of the highest refinement, requires an appreciation for all things and all arts, all aspects of life, for that matter, including religion. One who practices the occasional religious values for example the "retirement" ritual and leaving the world, is regarded with great appreciation. They are as aware as they are pious: they see the folly and shallowness in life and its struggle for miyabi and break free of it to pursue more heavenly goals.
The fact of the matter is, the practice of aware is when Buddhism and religion really come into play. An ideal countenance for the courtly figures is actually an appreciation and reverence for religion, but not the devoted practice of it. The tale of "The Woman Who Loved Insects" is a prime example of how society would view someone who truly practised what they preached. It's admirable, but weird in real life and gets you nowhere in society. Social status and the worldly life is now, and unfortunately religion in all practicality isn't very useful in an attempt to gain power.
A lack of religion, however, makes you seem shallow and unrefined, so an appreciation of it is demanded by society. Therefore, as seen in many cases especially in "Genji Monogatari", practices and rituals (not just of Buddhism but religion in general) such as a period of respect after an individuals death, performing acts for the purpose of generating karma, and the ideas of rebirth and the afterlife (a rather controversial idea in these texts; most seem to have a different idea of it) are still performed. It is not exactly like today's society for want of a little more superstition, perhaps, but in all honesty, religion is that nice little thing you perform in church every Sunday morning and talk about only in coffee houses or your deathbed.

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