The only explanation I've seen for its removal is that the Mexican government objected, but that doesn't seem like enough of an explanation. For example - why wasn't that a reason not to use it in the first place? And what leverage did Mexico have? As for the first part, I know there was a general climate of heightened racism / xenophobia in the 20s, but how did that lead to Mexicans / Mexican Americans being designated as distinct race? Or was there some other reason?
Another reader pointed me to this article, which I found really helpful (in case others are interested): https://neukom.dartmouth.edu/docs/16_gratton_la_raza_merchant.pdf
Gratton, B. and Klancher, E.M. 2016. "La Raza: Mexicans in the United States Census." Journal of Policy History, Vol. 28, No. 4, pp. 1-30.