This is handled in several ways – allowing favored nations to use up unused immigration quota from prior years, some labor mobility from the new Empire, a host of private immigration bills for family reunificiation. Family reunification provides yet another form of patronage for the Democratic urban machines. Your ward boss has to co-sign your request to bring in Cousin Giovanni…Also Presidential exemptions made for skills – scientists, technicians, special skills [German refugees who can help us duplicate their advanced optics for example].
This is turn presents the beginnings of a labor shortage in the Empire. This is handled by using the oncoming world crisis as a population gathering system. The test cases are the persecuted Iraqi Christians and Austrian Social Democrats from their mini civil war of 1934. By 1936 they have gathered in the Jews of Poland, the Baltic States, and Rumania plus many Arab Christians, Armenians, White Russians, and European political refugees. Those with key tech or scientific skills get exceptional leave to enter the CONUS. For the rest, the extra people are swept into Empire and the Base Area on special external only passports (what the Brits did to the East African expats in OTL)– all rights of US green card holder except right of entry to lower 48. Oh yes, they are also liable to US taxes and conscription.