I'd think that the technological hurdles would be relatively low, but not nonexistent as per VOY "Thirty Days". The operational rationale is no doubt going to be debated for quite some time, though.
The holotrap transport of ST:INS was hidden uncrewed and otherwise unattended, waiting for the conclusion of a rather monomaniacal mission. It's a good way to stash away something that's not needed for anything else and should go unobserved not only by natives but also by starfarers.
No doubt a big starship can also hide underwater waiting for the conclusion of a recce mission or whatever. But shouldn't NCC-1701 have better things to do? If her capabilities are not needed for the mission, and she's there merely to pick up Kirk at the end of an adventure, a shuttle could be used instead, hidden underwater, on a mountaintop, buried in sand, cloaked, whatever. Why land the ship?
I sort of doubt the movie will give us a rationale, such as transporters not being available and shuttles being undesirable for a clear-cut reason X. There's no big reason not to dip the ship, sure, but only if we postulate a short mission. If Kirk is gonna be gone for weeks, the ship should be free to depart.
But I might be dead wrong. Apparently, and spoilers be damned, there are plenty of things going on at the planet where this dipping takes place. A center of operations close to all
the action might be an operational necessity for a reason that becomes obvious once we see all the relevant scenes.