Here is what the Bible says in an issue similar to White Privilege:
James 1
2Consider it pure joy, my brothers and sisters,
a whenever you face trials of many kinds,
3because you know that the testing of your faith produces perseverance.
4Let perseverance finish its work so that you may be mature and complete, not lacking anything.
....................................
9Believers in humble circumstances ought to take pride in their high position.
10But the rich should take pride in their humiliation—since they will pass away like a wild flower.
11For the sun rises with scorching heat and withers the plant; its blossom falls and its beauty is destroyed. In the same way, the rich will fade away even while they go about their business.
So by application of God's word: For the faithful, trials produce spiritual maturity. Christians of low privilege have more trials and more opportunities to become more spiritually whole.
Black Christians (and all those in humble circumstances on Earth) should be proud because God actually sees them as highly privileged to suffer and become spiritually mature. This will lead to a larger eternal reward.
Whites (and all those of earthly high privilege) are actually seen by God as disposable. Even if faithful, their lesser suffering will lead to lesser spiritual maturity and presumably a lesser reward in heaven.
Caveat: God judges individuals and some Blacks, like President Obama, are more privileged than most whites. Likewise, some whites are less privileged than the average black person.