I think there's a simple solution, and it's to get the plaintiff's bar to do the work for the anti-immigration crowd.
Basically, you pass a federal law allowing the federal government to be sued for injuries people sustain from non-individually, case-by-case, vetted immigrants (negligent admission). The law would also allow the administrators to be held directly liable too, and not indemnified by the United States. The damages would be non-dischargeable in bankruptcy.
See the problem is that when immigration is restricted, there's always someone with standing to make a legal claim. When it's expanded, there isn't. So it's a one way ratchet. Creating a statutory cause in the other direction creates the necessary balance to fight the leftist administrations in the courts.