In my opinion, the Fourth Circuit pretty much has it dead to rights even without the tweets (which, from what I'm reading, they didn't consider anyway). I'm not versed enough in the legal technicalities of the President's national security powers nor what constitutes relative degrees of broadness re:country bans, but I do know about the Lemon test, which is what the Fourth used in their judgement.
It is obvious on its face that the travel ban/restriction/proclamation of glory/whatever is a functional problem for muslims, all the nations listed in the ban are muslim-majority and prohibits their travel to the United States even if it would otherwise be wholly legal and legitimate. Even if I believe, and I do not, that D.Trump and company did not intend to create a religious test to enter the United States (and Ban Mk.1's exemptions for non-Muslims show otherwise), they have created one in material reality.
So, can this infringement on religious identity be prohibited?. Lemon gives us three standards which must all be met.
1. The statute must have a secular legislative purpose. (Also known as the Purpose Prong)
2. The principal or primary effect of the statute must not advance nor inhibit religion. (Also known as the Effect Prong)
3. The statute must not result in an "excessive government entanglement" with religion. (Also known as the Entanglement Prong)
The Fourth found it in violation of the Purpose Prong, because a reasonable observer would be able to conclude even in spite of facial secularism that the purpose of the executive order was to bar individuals from the United States on the basis of religious belief. Secular motive perhaps, but non-secular purpose. They also questioned the Effect Prong but I lost my place in the ruling and couldn't find it again.
Regardless, they found that any temporal loss of First Amendment rights constitutes an irreparable harm, and thus if the executive order does this without justification it must be immediately given injunction.
I'm not sure if the question of "less restrictive measures" comes into play on this legal avenue or not, but if it does the Administration is straight fucked. You are
never going to win an argument that a group of blanket nationality bans are the
least restrictive thing you can do to stop terrorism.