Wouldn't it therefore be sensible for the WH to find another judge on another circuit who could issue a judgement declaring the Robart judgement to be unconstitutional on the above grounds and ordering State Department employees to submit to the original EO.
No because the U.S. legal system doesn't work that way. Any federal judge can impose a nationwide injunction and there's a process that plays out. The judiciary is extremely deferential, usually, to the executive and legislative branches and so any injunction is rare.
Once a judge issues that order, only an appeals court can override it, and since the 9th Appeals Court has refused to override it pending a full hearing (as I understand it), only the Supreme Court can override the appeals court. (Or maybe there's another step between, where the DC Court hears the appeal and then tracks it to the USSC).
Until then, the injunction stays in place. The Administration could appeal the Appeals Court and ask the Supreme Court for an emergency ruling, but I doubt that with the multiple suits making its way nationwide that the Supreme Court would short circuit the process. They'll likely amalgamate the various suits into one giant case (or take one particular case and say all similar cases will have the same ruling) and then hear it.
Kennedy (or Roberts) would be the deciding vote on this, and if Kennedy is still on the bench, he's got a long streak of being friendly to civil liberties and ruling against the Administration when it comes to cases like this (
Hamdi, 2006 comes to mind). Kennedy, in particular, may take umbrage at the Trump tweets and attacks on the judiciary and might rule against Trump in that scenario.
I should add that that the judiciary probably has no fear of Trump, given his popular vote loss and tenuous relationship with the Congressional Republicans. They read polls too and they probably feel very safe rejecting the President, knowing that there's a very vociferous opposition in place to Trump and Co. So the judiciary will continue telling Trump to pound sand if they feel like it.
This all means the injunction would probably end up staying in place until June (at the earliest) or if the justices decide they want to slow walk it, not until summer 2018. Most likely, the Supreme Court will add it to their docket and hear and issue a final ruling by June.
