I know VALUEH is by Housing weight and RACE is by Person weights - so how can you determine Housing value by Race? I’m not sure on whether this is possible given different weights.
The answer to the question about which weights you should use always depends on the level of your analysis.
You could perform this analysis at the household level. This would require you to either come up with some measure of the racial makeup of the individuals within each household or simply only look at the race of the first person listed on the census form (PERNUM==1). Since this will be household level analysis, you should use the household level sampling weight, HHWT.
Alternatively, you could perform this analysis at the individual level. Each person has their own unique race value and live within households with various values. Since this will be individual level analysis, you should use the individual level sampling weight, PERWT.
Both methods may be more or less well suited for different research questions.