It's easy
Well, your idea works and might steal everybody else's thunder.
Here's another idea-
1. Something like the National Origins Quota act is established no later than the end of the Civil War, possibly due to a perception of Irish-American American disloyalty (NY Draft Riots- their political alliance with Copperhead Democrats)- This remains in place until such a late point in the 20th century that Irish, German and Scandinavian migration is greatly reduced, immigration from southern and eastern Europe is shrunk to almost nothing. That eliminates a heap of the white population increase, and is amplified by the fact that immigrants tended to have larger families than the native born whites. It also cuts down on Chinese immigration and especially Japanese immigration, but I'll solve that later. What this law most certainly does is increase the share of Americans who are black.
2. Like OTL's National Origins immigration quota system, it does not have a limiting effect on Mexican migration. When Mexico develops to the point that its villagers are ready to emigrate, they do so in numbers at least as strong as in OTL.
3. There is a Spanish-American war around the same time as the historical timeframe, which IMHO is fairly likely. In addition to acquiring the Philippines, Guam and Puerto Rico, in the ATL the US annexes Cuba. Also, let's say that through the butterfly effect, the Spanish-American war occurs before Spain sells any islands to Germany, so the US gets the Carolines and Palaus and Marianas as part of the victory settlement.
4. At some point in the first quarter of the 20th century, the US gets into an alt-WWI, a great power war in Eurasia which requires it to mobilize.
With the draft and industrial mobilization, industry and the military suck up much of the agricultural labor force, black and white from the south. As in OTL's WWI, the US needs more laborers and infantrymen, and makes Puerto Ricans citizens with the right to immigrate to the mainland USA, and subject to conscription. With a smaller population base however, the USA applies the same laws to Cuba, Guam and the Philippines. Lots of guys from these countries join the infantry, and labor-starved agricultural enterprises get to employ many other in the sunbelt and points south. Filipino immigration happens to be encouraged in the ATL because for the duration of the war there is bitterness and suspicion against Mexicans in the southwest USA because of some pro-German activitties there and something like the Villa raid. The increased Filipino and Pacific Islander (linked w/ Asians by the US census) migration more than makes up for lost Japanese immigration, and Chinese immigration is a wash in OTL because, even though the national origins act restricts Chinese to a fairly small quota, the reduced flow prevents political pressure from ever causing a complete Chinese exclusion act.
With so many opportunities for laborers in the US, the flow of people from the Spanish War territories is truly massive, helping round out US demography to your desired proportions. The heroic participation of all these islanders strengthens the case for full statehood for Puerto Rico and Cuba especially, even if not the Philippines. The US economy, even if it has a post-WWI bust, soon goes into boom again and remains attractive to migrants from within US possessions. The advent of postwar radicalism in Europe prevents some suggestions by racists that the US needs to import more whites from Europe from being adopted.