Amnesty will will accelerate the evil of abortion. The only major political party that opposes abortion (tenuously) will become extinct with amnesty. Immigrants from Mexico and Central/South America vote for the party that promises them other people's money, which happens to be the party that promotes abortion. That is the connection.
An ancillary, or perhaps more intrinsic problem is the bishops' approval and promotion of statism, in this case government control of medical care. What they are in effect saying is, "Yes, I believe that the government should control every individual, (except us), and every organization, (except ours), and that the government should force every American citizen to comply with the demands of the Enlightened Government Elite, except for us.
A jackboot on the throats of everybody else, but give us an exemption. Now we hear whining because the jackboot makes no exceptions. Boo hoo.
That'll stop it cold because good, moral, people won't break the law and hire illegals, rent living space to illegals, or help them in any way.
Yeah, let's stick to anything but the real problem which is that the majority of the population is so hedonistic and immoral that they shrug off infanticide the same way Carthaginians sacrificed their children to Moloch, the god who they believed would grant them prosperity in return for their firstborn child.