Could SCOTUS rule that not only is it not allowed to ban the transportation of legal firearms you shouldn’t need multiple permits do to so this leading to Constitutional Carry and not needing a permit to transport a legally owned firearm?
Not likely. There's a reason the case they picked is at the
far end of the range of restrictions on firearms outside the home... constitutional carry is the far other end.
They don't need to reach any question of
carry whatsoever to strike down a law that prohibits locked-and-unloaded (or even disassembled) transportation. Carrying in a manner suitable for self-defense is one thing, merely hauling the gun to your other domicile in the Catskills or to a weekend pistol match in PA is quite another.
Moreover, even if they wanted to tackle carry right, NYC's carry scheme is basically that they
won't issue carry permits to ordinary people. Even if the Court takes a different case to deal with that (or decides to opine on it here, even though it's not at issue), the most you might realistically get is shall issue.
Here's the thing about "constitutional carry" - all
kinds of other constitutional rights can be subject to licensing schemes. Lots of public gatherings/protests require a permit, even though free speech and peaceable assembly are both protected by the constitution. Marriage is constitutionally protected - and requires a license from the state. Voting requires registration.
The RKBA has traditionally been treated
worse and taken
less seriously than most other constitutional rights. "Constitutional carry" being found to be constitutionally-required would mean that the Court suddenly jumped the RKBA
past other core rights in terms of how much the political branches are allowed to impose "reasonable" or "modest" licensing schemes. In short:
There is no realistic chance. That is not going to happen. 
