Short term punishment for states: ICE and the National Guard get sent into cities to make people feel unsafe, under the guise of an ‘immigration emergency’. Perhaps also Marines!
To punish more fully, just illegally withhold federal funds for whatever is most hurtful. Highways? Education? Healthcare?
And to your direct point, I’m sure someone could whip up a reason for the military to take over and shut down the sites if they don’t comply - this _is_ a national security matter after all.
Court system stops any of that? Just comply (or pretend to) with the letter of the ruling and try another barely-distinguishable but arguably different illegal method for the next few months while the gears of the court system grind.
What is _meant_ to stop the executive branch (meant to ‘execute’ the will of Congress, not just follow its own desires) going rogue is impeachment by Congress, but that seems like a far off prospect.