Which is why, if we’re smart, we’re leery about more immigration and want to keep it controlled.

Most people who are smart on immigration are actually not too exercised about regularizing current illegals, provided we have control of our borders first (meaning that we can largely control both physical access across the border and visa overstayers) and have a sensible ongoing immigration policy (one that does a couple of the following: restricts chain migration through ‘family reunification,’ favors skilled and/or wealthy immigrants over unskilled labor or lottery immigrants, favors compatibility with American culture, aims too prevent too large a concentration of immigrants from a distinct ethnic or other self-consciously separate background with the aim of promoting assimilation, and eliminates birthright citizenship tourism).
The future composition of the America people should be the choice of the American people.

I don’t see anything fundamentally incompatible between this point of view and the Church’s PRs on immigration, much less the gospel.