Were they able to avoid complying with discriminatory laws and customs because of their wealth and power, or were they subjected to the same treatment? Why were they exempted/not exempted? I am mainly interested in the post WWII period.
The Other Brahmins: Boston's Black Upper Class, 1750-1950 by Adelaide Cromwell addresses your question directly.
Bless her heart, Wikipedia says she is still with us at age 97.
IIRC, this book began as her PhD dissertation, then was slightly reworked into a book...that still reads like a dissertation.
Just a few points, but I suggest you read the whole thing:
The Black elite was economically what we today would call solidly middle class. There was never a 'leisure class' as she put it.
Holding income constant, they bough more TV's earlier, were members of formal social clubs (Elks lodge, Freemasons, college sorority alumni, etc.) in greater numbers, better cars, but held less real estate and of lower quality than white households of the same time/place/income. This was explained by their having less access to swanky restaurants, night life, etc.
College degree holding women working outside the home was more acceptable earlier. (The black female author with her PhD from the '40s and tenure-track career in a white college wrote with approval of this.)
You were 'upper class' based more on being from an old, prominent family than your wealth. If your family had been in Boston, and not been slaves, since the Revolutionary War, and you had a mediocre post office job, and your wife was an elementary school teacher in the black school, your photo in formal tuxedo at your daughter's debutante coming out ball had a photo in the book. You were upper class because of factors available to the black community. The book is really properly fascinating enough that it is making me ramble in this too-long post.
This one is not so good - black graduates from 'white' colleges looked down on graduates from HBCU's. Your daughter and her Smith College or Radcliff teaching degree and swanky sorority membership would have been too good for John Q. Tuskeegee grad.