Yes you can. democrats at the state level pull this in the courts all the time with Second Amendment stuff. All. The. Time.

No you can't. Remember Trump's so-called "Muslim Travel Ban"? There were actually two Executive Orders that he issued. After the first, the EO was enjoined by the court within five days. The second was enjoined by the court in eight days.

"Second Amendment stuff" can often stay on the books longer, because to get a temporary injunction you have to prove up that you have a reasonable chance of success on the merits. Here, we are (by construction) talking about an action that is clearly contrary to the law. The TRO hearing will be held within about a week of the policy being enacted.

You can't "buy time" this way.

If I wanted to smuggle somebody into the country, am I going to

A) Fly them into an airport, where they'll need to scan a valid passport at the source airport and pass through US customs when they get to the US?
B) Walk them across the border and have them throw their real passports into the Rio Grande when they get to the US?

Considering that both of the major terrorist attacks in the U.S. (the WTC bombing an 9/11) happened under Scenario A, that seems far more likely for someone that was trying to enter the country for the purposes of committing a terrorist act.

It's worth recalling that not even Trump tried this little maneuver. Again, when he adopted his travel bans, he had the sense at least to adopt the ban on everyone coming in from those countries - he named the nation as a threat, and then everyone was prohibited from coming in, whether they had a valid visa and passport or not. Because it's going to be nearly impossible to convince a judge that the threat from the migrant population claiming asylum is going to be significantly different from the threat profile of the population at large.

They already know of several terrorists they've caught along the border.

Did any of them claim asylum? Again, if you're a terrorist who wants to enter the country illegally, your best bet is to try to do it without being caught - and then if you're caught, to just accept deportation and try again. Formally applying for asylum and sitting down for additional background checks and interviews is beyond foolhardy.