I have no problem with class-based affirmative action, though I think it should supplement, not replace, its racial peer, as both are independent sources of disadvantage and both are independent sources of diversity. But even the premise is wrong here. The reason affirmative action inspires disdain isn't because it leaves "deserving" Whites out. The reason is simply that it is a program whose primary beneficiaries are seen to be Black people.
It is well established that even class-based programs which are tagged as predominantly helping Blacks have significant struggles maintaining their popularity. Welfare is a great example of this -- it is, of course, the paradigmatic class-based assistance program, and yet it is just as controversial and (more importantly) just as racialized as affirmative action is (Welfare Queens, anyone?). Affirmative action is particularly vulnerable to this, because many of the arguments deployed against it now work just as well against the class-based variety (that we should "just judge everyone based on merit" applies as much to the poor as it does to Blacks).
Advocates for class-based AA say that it would still help many Blacks, because Blacks are disproportionately concentrated amongst the extreme poor. But if that's the case, then I doubt you'll see any serious dissipation in the antipathy many Whites feel for affirmative action. The causal force is White jealousy of Black attainment. At the end of the day, the procedures don't matter all that much.
See also, Ta-Nehisi Coates: "There's always a good reason to be a racist."
"Sometimes I have feared that, in some wild paroxysm of rage, the white race, forgetful of the claims of humanity and the precepts of the Christian religion, will proceed to slaughter the Negro in wholesale, as some of that race have attempted to slaughter Chinamen, and as it has been done in detail in some districts of the Southern States. The grounds of this fear, however, have in some measure decreased since the Negro has largely disappeared from the arena of Southern politics, and has betaken himself to industrial pursuits and the acquisition of wealth and education, though even here, if over-prosperous, he is likely to excite a dangerous antagonism; for the white people do not easily tolerate the presence among them of a race more prosperous than themselves. The Negro as a poor ignorant creature does not contradict the race pride of the white race. He is more a source of amusement to that race than an object of resentment. Malignant resistance is augmented as he approaches the plane occupied by the white race, and yet I think that the resistance will gradually yield to the pressure of wealth, education, and high character.
My strongest conviction as to the future of the Negro therefore is, that he will not be expatriated nor annihilated, nor will he forever remain a separate and distinct race from the people around him, but that he will be absorbed assimilated, and will only appear finally, as the Phoenicians now appear on the shores of the Shannon, in the features of a blended race.
Frederick Douglass, "The Future of the Colored Race", in Negro Social and Political Thought: 1850-1920, Howard Brotz, ed. (New York: Basic Books 1966), pp. 308-310 (originally published May 1886), 309.