Delaying a re-entry application 3 months is hardly what passes for "unreasonable" in government circles. The fact that cases can still be reviewed and approved faster if necessary makes this even more reasonable legally.
> (e) Notwithstanding the temporary suspension imposed pursuant to subsection (a) of this section, the Secretaries of State and Homeland Security may jointly determine to admit individuals to the United States as refugees on a case-by-case basis, in their discretion, but only so long as they determine that the admission of such individuals as refugees is in the national interest -- including when the person is a religious minority in his country of nationality facing religious persecution, when admitting the person would enable the United States to conform its conduct to a preexisting international agreement, or when the person is already in transit and denying admission would cause undue hardship -- and it would not pose a risk to the security or welfare of the United States.
If the EO had merely delayed applications for three months pending additional procedures, courts across the country wouldn't be granting TROs to stop enforcement of it.
The Constitutional claims are very strong, at least as applied to certain groups of people affected by the EO: https://news.ycombinator.com/item?id=13531117.
You are conflating the right to enter the U.S. with due process rights relating to denial of U.S. entry. Only citizens have a right to enter the U.S. (i.e. they cannot be denied entry, period). But the Supreme Court has squarely held that permanent residents, at least, have a right to due process before being denied re-entry. That doesn't mean the government can't deny them re-entry, but it does mean that the government cannot do what it did here: deny them re-entry without individualized process (usually, a hearing).
The EO is going down. The only question is what parts of it will survive.