You can tie the aid to being the descendant of someone previously wronged by discrimination.
E.g., did your [great] grandfather serve in the military but get denied access to GI bill benefits since they lived in a state with segregated education -> you should be eligible for similar benefits.
E.g., did your parents/grandparents lived in an area that faced economic stagnation due to redlining / other policies which targeted neighborhoods -> you should eligible for aid/loans.
If laws/polices can be tailored to discriminate against black people without mentioning race, they can also be tailored to redress grievances without mentioning race. Tie the aid to being a victim (or descendant) of specific acts of discrimination. Sure there will be a few white people that qualify for the aid, but just like the white people hurt by the original discriminatory policies, they'll be a rounding error.