I am not sure what's so convoluted about a Church wanting to help people...
So let's address the elephant in the room here.
You keep pretending that the Catholic Church is somehow directly responsible for mass migration, when that simply is not the case. These Catholic NGOs you talk about have always maintained a distinction between people who are genuinely seeking asylum and those who are malicious bad actors taking advantage of the situation.
Here is an excerpt from a
recent article on the matter.
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The Church has no objection to you getting rid of the criminal element that you're complaining about, the concern is that you do not throw the baby out with the bathwater.
The same could be said about the State here. Church has an inherent mission and duty to help all people, but the State only has a duty to its citizens. Take the above example, the Church is morally obliged to speak out for those genuine refugees, but that does not mean the State has to listen.
Essentially what you're asking for is the Church to abandon its moral principles because the State is not doing it's job. What sense does that make?