Fundamental to citizenship is the right of self-identity and self-expression. Race-based classification of individuals deprives them of their self-identity and self-expression, and effectively their citizenship.
That argument does not seem to pertain to race-based self-classification in any way.
Ask the question in the Census, I suppose.
Prior to the 1870 Census, Congress considered asking that question on the census, but didn't come to agreement. A government official asked for local census takers to make inquiries as to the number of disenfranchised voters, which had spotty results, with some areas carefully reporting the number of illiterates and feeble-minded and other areas reporting at all.
When Congress was considering the apportionment in 1871/2, some representatives did the actual calculation and determined that it would have made no difference. Northern representatives were well aware of the effect of the removal of the 3/5 rule, and were disappointed that Southern states hadn't disenfranchised more voters. But they did stick the apportionment clause of the 14th Amendment into statute, where it remains today.
There were actually two apportionment bills after the 1870 Census. The second added a few representatives so no state lost representatives, but is not supportable on the basis of the actual population.
When the 14th Amendment was being considered, it was supposed to be based on voters (males over 18) but the New England states thought they would lose seats, because they had more children and relatively more women. People moving West were more likely to be men (multiple sons, or whose father was still living and owning the family farm in the East). So they switched to the rather awkward proportionality version.
Had they gone with the original version, it would be considered quite normal to ask citizenship status.